Search: a199296 -id:a199296
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A215703
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A(n,k) is the n-th derivative of f_k at x=1, and f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways; square array A(n,k), n>=0, k>=1, read by antidiagonals.
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+10
58
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1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 2, 12, 8, 0, 1, 1, 6, 9, 52, 10, 0, 1, 1, 4, 27, 32, 240, 54, 0, 1, 1, 2, 18, 156, 180, 1188, -42, 0, 1, 1, 2, 15, 100, 1110, 954, 6804, 944, 0, 1, 1, 8, 9, 80, 650, 8322, 6524, 38960, -5112, 0, 1, 1, 6, 48, 56, 590, 4908, 70098, 45016, 253296, 47160, 0
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OFFSET
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0,9
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COMMENTS
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A000081(m) distinct functions are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways. Some functions are representable in more than one way, the number of valid parenthesizations is A000108(m-1). The f_k are ordered, such that the number m of x's in f_k is a nondecreasing function of k. The exact ordering is defined by the algorithm below.
The list of functions f_1, f_2, ... begins:
| f_k : m : function (tree) : representation(s) : sequence |
+-----+---+------------------+--------------------------+----------+
| f_2 | 2 | x -> x^x | x^x | A005727 |
| f_3 | 3 | x -> x^(x*x) | (x^x)^x | A215524 |
| f_4 | 3 | x -> x^(x^x) | x^(x^x) | A179230 |
| f_5 | 4 | x -> x^(x*x*x) | ((x^x)^x)^x | A215704 |
| f_6 | 4 | x -> x^(x^x*x) | (x^x)^(x^x), (x^(x^x))^x | A215522 |
| f_7 | 4 | x -> x^(x^(x*x)) | x^((x^x)^x) | A215705 |
| f_8 | 4 | x -> x^(x^(x^x)) | x^(x^(x^x)) | A179405 |
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LINKS
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 2, 4, 2, 6, 4, 2, 2, ...
0, 3, 12, 9, 27, 18, 15, 9, ...
0, 8, 52, 32, 156, 100, 80, 56, ...
0, 10, 240, 180, 1110, 650, 590, 360, ...
0, 54, 1188, 954, 8322, 4908, 5034, 2934, ...
0, -42, 6804, 6524, 70098, 41090, 47110, 26054, ...
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MAPLE
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T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
end():
A:= (n, k)-> n!*coeff(series(subs(x=x+1, f(k)), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);
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MATHEMATICA
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T[n_] := If[n == 1, {x}, Map[x^#&, g[n - 1, n - 1]]];
g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten @ Table[ Table[ Table[ Product[T[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {v, g[n - i*j, i - 1]}], {w, Subsets[ Range[ Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];
f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];
A[n_, k_] := n! * SeriesCoefficient[f[k] /. x -> x+1, {x, 0, n}];
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CROSSREFS
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Columns k=1-17, 37 give: A019590, A005727, A215524, A179230, A215704, A215522, A215705, A179405, A215706, A215707, A215708, A215709, A215691, A215710, A215643, A215629, A179505, A211205.
Rows n=0+1, 2-10 give: A000012, A215841, A215842, A215834, A215835, A215836, A215837, A215838, A215839, A215840.
Cf. A000081, A000108, A033917, A211192, A214569, A214570, A214571, A216041, A216281, A216349, A216350, A216351, A216368, A222379, A222380, A277537, A306679, A306710, A306726.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A199085
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Number of distinct values taken by 3rd derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
10
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1, 1, 2, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 64, 74, 84, 95, 107, 119, 132, 146, 160, 175, 191, 207, 224, 242, 260, 279, 299, 319, 340, 362, 384, 407, 431, 455, 480, 506, 532, 559, 587, 615, 644, 674, 704, 735, 767, 799, 832, 866, 900, 935, 971, 1007, 1044
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OFFSET
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1,3
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COMMENTS
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Number of distinct values taken by 0th and 1st derivative is 1,1,1,1,1,1,... and by 2nd is 1,1,2,3,4,5,...
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LINKS
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FORMULA
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Conjectured g.f.: (x^6-2*x^4-x^2+x-1)*x/(x^5-2*x^4+x^3-x^2+2*x-1). - Alois P. Heinz, Nov 02 2011
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EXAMPLE
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For n=5 there are 7 distinct values: 9, 15, 18, 21, 24, 33, 48, they are given by 3rd derivatives of the following parenthesizations at x=1: x^(x^((x^x)^x)), x^((x^(x^x))^x), (x^x)^(x^(x^x)), x^(((x^x)^x)^x), (x^(x^x))^(x^x), (((x^(x^x)))^x)^x, (((x^x)^x)^x)^x. So a(5)=7.
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MAPLE
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f:= proc(n) option remember;
`if`(n=1, {[0, 0]}, {seq(seq(seq([2+g[1], 3*(1+g[1]+h[1])+g[2]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[2], f(n))):
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MATHEMATICA
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f[1] = {x}; f[n_] := Flatten[Table[Outer[Power, f[k], f[n-k]], {k, n-1}]]; Table[Length[Union[D[f[n], {x, 3}] /. x -> 1]], {n, 1, 8}] (* Reshetnikov *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A199205
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Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
10
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1, 1, 2, 4, 9, 17, 30, 50, 77, 113, 156, 212, 279, 355, 447, 560, 684, 822, 985, 1171, 1375, 1599, 1856, 2134, 2445, 2769, 3125, 3519, 3939, 4376, 4857, 5372, 5914, 6484, 7083, 7717, 8411, 9130, 9882, 10683, 11524, 12393
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OFFSET
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1,3
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LINKS
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EXAMPLE
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a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.
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MAPLE
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f:= proc(n) option remember;
`if`(n=1, {[0, 0, 0]},
{seq(seq(seq( [2+g[1], 3*(1 +g[1] +h[1]) +g[2],
8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[3], f(n))):
seq(a(n), n=1..20);
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MATHEMATICA
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f[n_] := f[n] = If[n == 1, {{0, 0, 0}}, Union @ Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
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CROSSREFS
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Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215834. Column k=4 of A216368.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A216368
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Number T(n,k) of distinct values taken by k-th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
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+10
10
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1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 7, 9, 9, 1, 5, 11, 17, 20, 20, 1, 6, 15, 30, 45, 48, 48, 1, 7, 20, 50, 92, 113, 115, 115, 1, 8, 26, 77, 182, 262, 283, 286, 286, 1, 9, 32, 113, 342, 591, 691, 717, 719, 719, 1, 10, 39, 156, 601, 1263, 1681, 1815, 1838, 1842, 1842
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OFFSET
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1,5
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COMMENTS
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T(n,k) <= A000081(n) because there are only A000081(n) different functions that can be represented with n x's.
It is not true that T(n,n) = T(n,n-1) for all n>1: T(13,13) - T(13,12) = 12486 - 12485 = 1.
Conjecture: T(n,n) = A000081(n) for n>=1. It would be nice to have a proof (or a disproof if the conjecture is wrong).
I made a descendant graph (Plot 1) that shows how each derivative relates to the next. In this picture the number of nodes in row k gives the value T(n,k). You can see at n=6 collisions begin to occur, and at n=7 the situation is even worse. I then computed a new triangle with collisions removed (Plot 2) and values:
1
1 1
1 2 2
1 3 4 4
1 4 7 9 9
1 5 11 88 20 20
1 6 16 34 46 48 48
I suspect that Plot 2 will admit a recursive construction more readily than the graphs with collisions. You can already see that each graph "n-1" is a subgraph of graph "n" and that the remainder of graph "n" is similar to graph "n-1" with additional branches. (End)
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LINKS
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EXAMPLE
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For n = 4 there are A000108(3) = 5 possible parenthesizations of x^x^x^x: [x^(x^(x^x)), x^((x^x)^x), (x^(x^x))^x, (x^x)^(x^x), ((x^x)^x)^x]. The first, second, third, fourth derivatives at x=1 are [1,1,1,1,1], [2,2,4,4,6], [9,15,18,18,27], [56,80,100,100,156] => row 4 = [1,3,4,4].
Triangle T(n,k) begins:
1;
1, 1;
1, 2, 2;
1, 3, 4, 4;
1, 4, 7, 9, 9;
1, 5, 11, 17, 20, 20;
1, 6, 15, 30, 45, 48, 48;
1, 7, 20, 50, 92, 113, 115, 115;
...
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MAPLE
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with(combinat):
F:= proc(n) F(n):=`if`(n<2, [(x+1)$n], map(h->(x+1)^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [(x+1)^n],
`if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
end:
T:= proc(n) local i, l;
l:= map(f->[seq(i!*coeff(series(f, x, n+1), x, i), i=1..n)], F(n));
seq(nops({map(x->x[i], l)[]}), i=1..n)
end:
seq(T(n), n=1..10);
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MATHEMATICA
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g[n_, i_] := g[n, i] = If[i==1, {x^n}, Flatten@Table[Table[Table[Product[ T[i][[w[[t]] - t+1]], {t, 1, j}]*v, {v, g[n - i*j, i-1]}], {w, Subsets[ Range[Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];
T[n_] := T[n] = If[n==1, {x}, x^#& /@ g[n-1, n-1]];
T[n_, k_] := Union[k! (SeriesCoefficient[#, {x, 0, k}]& /@ (T[n] /. x -> x+1))] // Length;
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CROSSREFS
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Main diagonal gives (conjectured): A000081.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A199883
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Number of distinct values taken by 6th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
7
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1, 1, 2, 4, 9, 20, 48, 113, 262, 591, 1263, 2505, 4764, 8479, 14285, 22871, 35316, 52755, 76517, 107826, 148914, 202715, 270622
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OFFSET
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1,3
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LINKS
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EXAMPLE
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a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 6th derivative at x=1: (x^(x^(x^x))) -> 2934; ((x^x)^(x^x)), ((x^(x^x))^x) -> 4908; (x^((x^x)^x)) -> 5034; (((x^x)^x)^x) -> 8322.
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MAPLE
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f:= proc(n) option remember;
`if`(n=1, {[0, 0, 0, 0, 0]},
{seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
+15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4],
45*h[1]*g[1]^2+(120+60*h[2]+15*h[3]+60*g[2]+
270*h[1])*g[1]+54+15*h[3]+30*g[3]+6*g[4]+
60*h[1]*g[2]+15*h[1]*g[3]+30*h[1]+ 20*h[2]*g[2]+
100*h[2]+90*h[1]^2+g[5]+60*g[2]+6*h[4]],
h=f(n-j)), g=f(j)), j=1..n-1)})
end:
a:= n-> nops(map(x-> x[5], f(n))):
seq(a(n), n=1..15);
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CROSSREFS
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Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215836. Column k=6 of A216368.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A215796
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Number of distinct values taken by 7th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
3
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1, 1, 2, 4, 9, 20, 48, 115, 283, 691, 1681, 3988, 9241, 20681, 44217, 89644
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OFFSET
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1,3
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LINKS
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EXAMPLE
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a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 7th derivative at x=1: (x^(x^(x^x))) -> 26054; ((x^x)^(x^x)), ((x^(x^x))^x) -> 41090; (x^((x^x)^x)) -> 47110; (((x^x)^x)^x) -> 70098.
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MAPLE
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T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
end():
a:= n-> nops({map(f-> 7!*coeff(series(subs(x=x+1, f), x, 8), x, 7), T(n))[]}):
seq(a(n), n=1..12);
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CROSSREFS
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Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215837.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A215835
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Fifth derivative of f_n at x=1, where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.
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+10
3
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0, 10, 240, 180, 1110, 650, 590, 360, 3160, 1880, 1180, 1420, 950, 1360, 890, 660, 480, 7050, 4410, 2770, 3130, 2300, 2070, 1480, 2670, 1840, 1370, 1070, 2610, 1780, 1190, 1310, 1010, 1080, 780, 600, 480, 13560, 8900, 5780, 3780, 6260, 4950, 4140, 3190, 3080
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OFFSET
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1,2
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COMMENTS
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For the ordering of functions f_n see A215703.
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LINKS
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MAPLE
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T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
end():
a:= n-> 5!*coeff(series(subs(x=x+1, f(n)), x, 6), x, 5):
seq(a(n), n=1..100);
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CROSSREFS
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Number of distinct values of a(n) taken for functions with m x's: A199296.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A215971
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Number of distinct values taken by 8th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
2
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1, 1, 2, 4, 9, 20, 48, 115, 286, 717, 1815, 4574, 11505, 28546, 69705, 166010
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OFFSET
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1,3
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LINKS
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EXAMPLE
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a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 8th derivative at x=1: (x^(x^(x^x))) -> 269128; ((x^x)^(x^x)), ((x^(x^x))^x) -> 382520; (x^((x^x)^x)) -> 511216; (((x^x)^x)^x) -> 646272.
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MAPLE
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# load programs from A215703, then:
a:= n-> nops({map(f-> 8!*coeff(series(subs(x=x+1, f),
x, 9), x, 8), T(n))[]}):
seq(a(n), n=1..10);
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CROSSREFS
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Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215838. Column k=8 of A216368.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A216062
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Number of distinct values taken by 9th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
2
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1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1838, 4734, 12247, 31617, 81208
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OFFSET
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1,3
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COMMENTS
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a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 9th derivative at x=1: (x^(x^(x^x))) -> 3010680; ((x^x)^(x^x)), ((x^(x^x))^x) -> 3863808; (x^((x^x)^x)) -> 6019416; (((x^x)^x)^x) -> 6333336.
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LINKS
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MAPLE
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# load programs from A215703, then:
a:= n-> nops({map(f-> 9!*coeff(series(subs(x=x+1, f),
x, 10), x, 9), T(n))[]}):
seq(a(n), n=1..11);
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CROSSREFS
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Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215839. Column k=9 of A216368.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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A216403
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Number of distinct values taken by 10th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.
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+10
2
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1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4763, 12452, 32711, 86239
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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LINKS
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EXAMPLE
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a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 10th derivative at x=1: (x^(x^(x^x))) -> 37616880; ((x^x)^(x^x)), ((x^(x^x))^x) -> 42409440; (x^((x^x)^x)) -> 77899320; (((x^x)^x)^x) -> 66712680.
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MAPLE
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# load programs from A215703, then:
a:= n-> nops({map(f-> 10!*coeff(series(subs(x=x+1, f),
x, 11), x, 10), T(n))[]}):
seq(a(n), n=1..11);
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CROSSREFS
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Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A199296 (5th derivatives), A199883 (6th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215840. Column k=10 of A216368.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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