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A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)*arcsin(x)) + sqrt(y)*sinh(sqrt(y)*arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k *x^n/n!.

%I A091885 #31 Jan 14 2025 01:48:49
%S A091885 1,1,1,1,4,1,9,10,1,64,20,1,225,259,35,1,2304,784,56,1,11025,12916,
%T A091885 1974,84,1,147456,52480,4368,120,1,893025,1057221,172810,8778,165,1,
%U A091885 14745600,5395456,489280,16368,220,1,108056025,128816766,21967231,1234948,28743
%N A091885 Triangle T(n,k) defined by the generating function cosh(sqrt(y)*arcsin(x)) + sqrt(y)*sinh(sqrt(y)*arcsin(x)) - 1 = Sum_{n>=1} Sum_{k=1..n} T(n,k)*y^k *x^n/n!.
%C A091885 Row sums are equal to A006228(n). This is sequence A121408 without the intertwining zeros. - _Emeric Deutsch_, Jul 28 2006
%C A091885 This number triangle corresponds to the coefficients of the polynomial of the denominator of Fourier cosine coefficients for functions of the form sin(x)^(2*k) for integer n. For example (k=5), evaluating Integral_{x=-Pi..Pi} cos(n*x)*sin(x)^10 dx, we have -7257600*sin(n*Pi)/(-14745600*n + 5395456*n^3 - 489280*n^5 + 16368*n^7 - 220*n^9 + n^11); note the sequence of the coefficients of the polynomial of the denominator: -14745600, 5395456, -489280, 16368, -220, 1. - _John M. Campbell_, May 28 2011
%F A091885 E.g.f.: cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1.
%e A091885 Triangle starts:
%e A091885     1;
%e A091885     1;
%e A091885     1,   1;
%e A091885     4,   1;
%e A091885     9,  10,   1;
%e A091885    64,  20,   1;
%e A091885   225, 259,  35,   1;
%p A091885 G:=cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1: Gser:=simplify(series(G,x=0,15)): for n from 1 to 13 do P[n]:=sort(expand(n!*coeff(Gser,x,n))) od: for n from 1 to 13 do seq(coeff(P[n],y,k),k=1..ceil(n/2)) od; # yields sequence in triangular form # _Emeric Deutsch_, Jul 28 2006
%t A091885 m = 14; (* number of rows *)
%t A091885 T = Rest /@ Rest[CoefficientList[#, y]& /@ (CoefficientList[Cosh[Sqrt[y]* ArcSin[x]] + Sqrt[y]*Sinh[Sqrt[y]*ArcSin[x]] - 1  + O[x]^(m + 1), x]* Range[0, m]! // Simplify[#, y > 0]&)];
%t A091885 Flatten[T] (* _Jean-François Alcover_, Sep 27 2021 *)
%Y A091885 Cf. A006228.
%Y A091885 Cf. A121408.
%K A091885 nonn,tabf,easy
%O A091885 1,5
%A A091885 _Karol A. Penson_, Feb 08 2004
%E A091885 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004