Search: a071952 -id:a071952
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A071951
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Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.
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+10
23
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1, 2, 1, 4, 8, 1, 8, 52, 20, 1, 16, 320, 292, 40, 1, 32, 1936, 3824, 1092, 70, 1, 64, 11648, 47824, 25664, 3192, 112, 1, 128, 69952, 585536, 561104, 121424, 7896, 168, 1, 256, 419840, 7096384, 11807616, 4203824, 453056, 17304, 240, 1, 512, 2519296, 85576448, 243248704, 137922336, 23232176, 1422080, 34584, 330, 1
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OFFSET
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1,2
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COMMENTS
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Removing a factor of 2^m from the m-th subdiagonal (the main diagonal corresponds to m = 0) gives the triangle A080248. - Peter Bala, Oct 15 2023
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LINKS
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G. E. Andrews, W. Gawronski and L. L. Littlejohn, The Legendre-Stirling Numbers, Discrete Mathematics, Volume 311, Issue 14, 28 July 2011, Pages 1255-1272.
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FORMULA
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T(n, j) = Sum_{r=1..j} (-1)^(r+j)*(2*r+1)*(r^2+r)^n/((r+j+1)!*(j-r)!).
G.f. for j-th column (without leading zeros): 1/Product_{r=1..j} (1 - r*(r+1)*x), j >= 1. From eq.(4.5) of the Everitt et al. paper.
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EXAMPLE
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The triangle begins:
n\j 1 2 3 4 5 6 7 8 9 ...
1: 1
2: 2 1
3: 4 8 1
4: 8 52 20 1
5: 16 320 292 40 1
6: 32 1936 3824 1092 70 1
7: 64 11648 47824 25664 3192 112 1
8: 128 69952 585536 561104 121424 7896 168 1
9: 256 419840 7096384 11807616 4203824 453056 17304 240 1
...
Row 10: 512 2519296 85576448 243248704 137922336 23232176 1422080 34584 330 1. Reformatted by Wolfdieter Lang, Apr 10 2013
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MAPLE
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N:= 20: # to get the first N rows, flattened
for j from 1 to N do S[j]:= series(x^j/mul(1-r*(r+1)*x, r=1..j), x, N+1) od:
seq(seq(coeff(S[j], x, i), j=1..i), i=1..N); # Robert Israel, Dec 03 2015
# alternative
option remember;
if k =0 then
if n = 0 then
1;
else
0;
end if;
elif n = 0 then
if k =0 then
1;
else
0;
end if;
else
procname(n-1, k-1)+k*(k+1)*procname(n-1, k) ;
end if;
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MATHEMATICA
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Flatten[ Table[ Sum[(-1)^{r + j}(2r + 1)(r^2 + r)^n/((r + j + 1)!(j - r)!), {r, j}], {n, 10}, {j, n}]]
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PROG
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(PARI) {T(n, k) = sum( i=0, k, (-1)^(i+k) * (2*i + 1) * (i*i + i)^n / (k-i)! / (k+i+1)! )} /* Michael Somos, Feb 25 2012 */
(Magma) [[(&+[(-1)^(r+j)*(2*r+1)*(r^2+r)^n/(Factorial(r+j+1)*Factorial(j-r)): r in [1..j]]): j in [1..n]]: n in [1..12]]; // G. C. Greubel, Mar 16 2019
(Sage) [[sum( (-1)^(r+j)*(2*r+1)*(r^2+r)^n/(factorial(r+j+1)*factorial(j-r)) for r in (1..j)) for j in (1..n)] for n in (1..12)] # G. C. Greubel, Mar 16 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A089272
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Fourth column (k=5) of array A078739(n,k) ((2,2)- generalized Stirling2) divided by 12.
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+10
3
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1, 48, 1412, 34400, 766416, 16296448, 337709632, 6896540160, 139644851456, 2813500878848, 56517475402752, 1133320271749120, 22702062218039296, 454469171469877248, 9094518828981174272, 181952003020274401280
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OFFSET
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0,2
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COMMENTS
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The numerator of the g.f. is the n=2 row polynomial of the triangle A089276.
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REFERENCES
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P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003), 198-205.
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LINKS
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FORMULA
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G.f. (1+8*x)/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)).
a(n)= (3500*20^n - 3780*12^n + 945*6^n - 35*2^n)/630 = d(n) + 8*d(n-1), with d(n) := A071952(n+4)= (2500*20^n - 2268*12^n + 405*6^n - 7*2^n)/630, n>=1.
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MATHEMATICA
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LinearRecurrence[{40, -508, 2304, -2880}, {1, 48, 1412, 34400}, 16] (* Jean-François Alcover, Feb 28 2020 *) (* Jean-François Alcover, Feb 28 2020 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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1, 70, 3192, 121424, 4203824, 137922336, 4380918784, 136378114048, 4191383868672, 127754693361152, 3873052857829376, 117001609550671872, 3526270158211870720, 106112798944292282368, 3189880933574260359168
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: 1/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)*(1-6*5*x)).
a(n) = (16875*(6*5)^n -20000*(5*4)^n +6048*(4*3)^n -405*(3*2)^n +2*(2*1)^n)/2520.
a(n) = det(|ps(i+5,j+4)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). [Mircea Merca, Apr 06 2013]
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PROG
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(Magma) [(16875*(6*5)^n - 20000*(5*4)^n + 6048*(4*3)^n - 405*(3*2)^n + 2*(2*1)^n)/2520: n in [0..20]]; // Vincenzo Librandi, Sep 02 2011
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A093050
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Exponent of 2 in (3^n-3)*2^(n-1).
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+10
2
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0, 0, 3, 2, 6, 4, 7, 6, 11, 8, 11, 10, 14, 12, 15, 14, 20, 16, 19, 18, 22, 20, 23, 22, 27, 24, 27, 26, 30, 28, 31, 30, 37, 32, 35, 34, 38, 36, 39, 38, 43, 40, 43, 42, 46, 44, 47, 46, 52, 48, 51, 50, 54, 52, 55, 54, 59, 56, 59, 58, 62, 60, 63, 62, 70, 64, 67, 66, 70
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OFFSET
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0,3
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LINKS
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FORMULA
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Recurrence: a(2n) = a(n) + [(n+1)/2] + 1, a(2n+1) = 2n.
G.f.: sum(k>=0, t^2(3+2t+2t^3-t^4)/[(1+t^2)(1-t^2)^2], t=x^2^k).
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PROG
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(PARI) a(n)=if(n<1, 0, if(n%2==0, a(n/2)+2*floor((n+2)/4)+1, n-1))
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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4, 52, 292, 1092, 3192, 7896, 17304, 34584, 64284, 112684, 188188, 301756, 467376, 702576, 1028976, 1472880, 2065908, 2845668, 3856468, 5150068, 6786472, 8834760, 11373960, 14493960, 18296460, 22895964, 28420812, 35014252
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OFFSET
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3,1
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LINKS
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FORMULA
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a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90.
a(0)=4, a(1)=52, a(2)=292, a(3)=1092, a(4)=3192, a(5)=7896, a(6)=17304, a(n)=7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7). - Harvey P. Dale, Jul 03 2011
E.g.f.: x^3*(60 + 135*x + 54*x^2 + 5*x^3)*exp(x)/90. - G. C. Greubel, Mar 16 2019
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MATHEMATICA
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Flatten[ Table[ Sum[(-1)^{r + n - 2}(2r + 1)(r^2 + r)^n/((r + n - 1)!(n - 2 - r)!), {r, 1, n - 2}], {n, 3, 34}]]
Table[(n-2)(n-1)n(n+1)(5n^2-11n+3)/90, {n, 3, 30}] (* or *) LinearRecurrence[ {7, -21, 35, -35, 21, -7, 1}, {4, 52, 292, 1092, 3192, 7896, 17304}, 30] (* Harvey P. Dale, Jul 03 2011 *)
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PROG
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(PARI) {a(n) = (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90}; \\ G. C. Greubel, Mar 16 2019
(Magma) [(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90: n in [3..30]]; // G. C. Greubel, Mar 16 2019
(Sage) [(n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90 for n in (3..30)] # G. C. Greubel, Mar 16 2019
(GAP) List([3..30], n-> (n-2)*(n-1)*n*(n+1)*(5*n^2 - 11*n + 3)/90); # G. C. Greubel, Mar 16 2019
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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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