%I #40 Oct 30 2023 09:52:50
%S 1,1,1,1,10,1,1,91,35,1,1,820,966,84,1,1,7381,24970,5082,165,1,1,
%T 66430,631631,273988,18447,286,1,1,597871,15857205,14057043,1768195,
%U 53053,455,1,1,5380840,397027996,704652312,157280838,8187608,129948,680,1
%N Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k).
%C This is table 4 on page 12 of Gelineau and Zeng, read downwards by columns.
%C Reversing rows gives A008958.
%C Apparently the table can also be obtained by deleting each second row and column of A136630.
%H Michael De Vlieger, <a href="/A160562/b160562.txt">Table of n, a(n) for n = 0..11475</a> (rows n = 0..150, flattened)
%H Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, <a href="https://arxiv.org/abs/2202.13978">Alternating runs of permutations and the central factorial numbers</a>, arXiv:2202.13978 [math.CO], 2022.
%H Yoann Gelineau and Jiang Zeng, <a href="http://arxiv.org/abs/0905.2899">Combinatorial Interpretations of the Jacobi-Stirling Numbers</a>, arXiv:0905.2899 [math.CO], May 18 2009.
%F 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
%F 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
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 10, 1;
%e 1, 91, 35, 1;
%e 1, 820, 966, 84, 1;
%e ...
%p 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
%t T[n_, k_] := Sum[(-1)^(k - m)*(2m + 1)^(2n + 1)*Binomial[2k, k + m]/(k + m + 1), {m, 0, k}]/(4^k*(2k)!);
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 22 2017 *)
%Y Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1), A185375 (right column k=n-2).
%Y Cf. A001819, A008275, A008277, A008958, A039599, A136630.
%K nonn,tabl
%O 0,5
%A _Jonathan Vos Post_, May 19 2009
%E More terms from _R. J. Mathar_, Sep 09 2009