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A360693
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Number T(n,k) of sets of n words of length n over binary alphabet where the first letter occurs k times; triangle T(n,k), n>=0, n-signum(n)<=k<=n*(n-1)+signum(n), read by rows.
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5
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1, 1, 1, 2, 2, 2, 3, 10, 15, 15, 10, 3, 4, 37, 108, 228, 336, 394, 336, 228, 108, 37, 4, 5, 101, 600, 2150, 5645, 11680, 19752, 27820, 32935, 32935, 27820, 19752, 11680, 5645, 2150, 600, 101, 5, 6, 226, 2490, 14745, 61770, 200529, 535674, 1211485, 2368200
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OFFSET
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0,4
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COMMENTS
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T(n,k) is defined for all n >= 0 and k >= 0. The triangle contains only the positive elements.
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LINKS
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FORMULA
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T(n,k) = T(n,n^2-k).
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EXAMPLE
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T(2,3) = 2: {aa,ab}, {aa,ba}.
T(3,3) = 10: {aab,abb,bbb}, {aab,bab,bbb}, {aab,bba,bbb}, {aba,abb,bbb}, {aba,bab,bbb}, {aba,bba,bbb}, {abb,baa,bbb}, {abb,bab,bba}, {baa,bab,bbb}, {baa,bba,bbb}.
T(4,3) = 4: {abbb,babb,bbab,bbbb}, {abbb,babb,bbba,bbbb}, {abbb,bbab,bbba,bbbb}, {babb,bbab,bbba,bbbb}.
Triangle T(n,k) begins:
1;
1, 1;
. 2, 2, 2;
. . 3, 10, 15, 15, 10, 3;
. . . 4, 37, 108, 228, 336, 394, 336, 228, 108, 37, 4;
. . . . 5, 101, 600, 2150, 5645, 11680, 19752, 27820, 32935, 32935, ...;
...
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MAPLE
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g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
g(n, i-1, j-k)*x^(i*k)*binomial(binomial(n, i), k), k=0..j))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=n-signum(n)..n*(n-1)+signum(n)))(g(n$3)):
seq(T(n), n=0..6);
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MATHEMATICA
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g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Binomial[n, i], k], {k, 0, j}]]]];
T[n_] := Table[Coefficient[#, x, i], {i, n - Sign[n], n(n - 1) + Sign[n]}]&[g[n, n, n]];
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CROSSREFS
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Main diagonal T(n,n) gives A154323(n-1) for n>=1.
T(n,n-1) gives A000027(n) for n>=1.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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