# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a160562 Showing 1-1 of 1 %I A160562 #40 Oct 30 2023 09:52:50 %S A160562 1,1,1,1,10,1,1,91,35,1,1,820,966,84,1,1,7381,24970,5082,165,1,1, %T A160562 66430,631631,273988,18447,286,1,1,597871,15857205,14057043,1768195, %U A160562 53053,455,1,1,5380840,397027996,704652312,157280838,8187608,129948,680,1 %N A160562 Triangle of scaled central factorial numbers, T(n,k) = A008958(n,n-k). %C A160562 This is table 4 on page 12 of Gelineau and Zeng, read downwards by columns. %C A160562 Reversing rows gives A008958. %C A160562 Apparently the table can also be obtained by deleting each second row and column of A136630. %H A160562 Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened) %H A160562 Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022. %H A160562 Yoann Gelineau and Jiang Zeng, Combinatorial Interpretations of the Jacobi-Stirling Numbers, arXiv:0905.2899 [math.CO], May 18 2009. %F A160562 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 A160562 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 A160562 Triangle starts: %e A160562 1; %e A160562 1, 1; %e A160562 1, 10, 1; %e A160562 1, 91, 35, 1; %e A160562 1, 820, 966, 84, 1; %e A160562 ... %p A160562 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 A160562 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 A160562 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 22 2017 *) %Y A160562 Cf. A002452 (column k=1), A002453 (column k=2), A000447 (right column k=n-1), A185375 (right column k=n-2). %Y A160562 Cf. A001819, A008275, A008277, A008958, A039599, A136630. %K A160562 nonn,tabl %O A160562 0,5 %A A160562 _Jonathan Vos Post_, May 19 2009 %E A160562 More terms from _R. J. Mathar_, Sep 09 2009 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE