OFFSET
0,5
COMMENTS
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
KEYWORD
nonn,tabl
AUTHOR
Jonathan Vos Post, May 19 2009
EXTENSIONS
More terms from R. J. Mathar, Sep 09 2009
STATUS
approved