A160562 Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).

1, 1, 1, 1, 10, 1, 1, 91, 35, 1, 1, 820, 966, 84, 1, 1, 7381, 24970, 5082, 165, 1, 1, 66430, 631631, 273988, 18447, 286, 1, 1, 597871, 15857205, 14057043, 1768195, 53053, 455, 1, 1, 5380840, 397027996, 704652312, 157280838, 8187608, 129948, 680, 1

Offset

0,5

Comments

This is table 4 on page 12 of Gelineau and Zeng, read downwards by columns.

Reversing rows gives A008958.

Apparently the table can also be obtained by deleting each second row and column of A136630.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened)

Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022.

Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 18 2009.

Formula

T(n,k) = (1/((2*k)!*4^k)) * Sum_{m=0..k} (-1)^(k-m)*A039599(k,m)*(2*m+1)^(2*n). - Werner Schulte, Nov 01 2015

T(n,k) = ((-1)^(n-k)*(2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sin(x)^(2*k+1) = ((2*n+1)!/(2*k+1)!) * [x^(2*n+1)]sinh(x)^(2*k+1). Note that sin(x)^(2*k+1) = (Sum_{i=0..k} (-1)^i*binomial(2*k+1,k-i)*sin((2*i+1)*x))/(2^(2*k)). - Jianing Song, Oct 29 2023

Example

Triangle starts:
  1;
  1, 1;
  1, 10, 1;
  1, 91, 35, 1;
  1, 820, 966, 84, 1;
  ...

Maple

A160562 := proc(n, k) npr := 2*n+1 ; kpr := 2*k+1 ; sinh(t*sinh(x)) ; npr!*coeftayl(%, x=0, npr) ; coeftayl(%, t=0, kpr) ; end: seq(seq(A160562(n, k), k=0..n), n=0..15) ; # R. J. Mathar, Sep 09 2009

Mathematica

T[n_, k_] := Sum[(-1)^(k - m)*(2m + 1)^(2n + 1)*Binomial[2k, k + m]/(k + m + 1), {m, 0, k}]/(4^k*(2k)!);
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2017 *)

Crossrefs

Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1), A185375 (right column k=n-2).

Cf. A001819, A008275, A008277, A008958, A039599, A136630.

Sequence in context: A176021 A166972 A364071 * A176243 A022173 A158117

Adjacent sequences: A160559 A160560 A160561 * A160563 A160564 A160565

Keyword

nonn,tabl

Author

Jonathan Vos Post, May 19 2009

Extensions

More terms from R. J. Mathar, Sep 09 2009

Status

approved