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!.

1, 1, 1, 1, 4, 1, 9, 10, 1, 64, 20, 1, 225, 259, 35, 1, 2304, 784, 56, 1, 11025, 12916, 1974, 84, 1, 147456, 52480, 4368, 120, 1, 893025, 1057221, 172810, 8778, 165, 1, 14745600, 5395456, 489280, 16368, 220, 1, 108056025, 128816766, 21967231, 1234948, 28743

Offset: 1

Author

Karol A. Penson, Feb 08 2004

Keywords

• nonn
• tabf
• easy

Comments

Row sums are equal to A006228(n). This is sequence A121408 without the intertwining zeros. - Emeric Deutsch, Jul 28 2006

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

Examples

			Triangle starts:
    1;
    1;
    1,   1;
    4,   1;
    9,  10,   1;
   64,  20,   1;
  225, 259,  35,   1;
		

Crossrefs

Cf. A006228.

Cf. A121408.

Programs

Maple

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

Mathematica

m = 14; (* number of rows *)
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]&)];
Flatten[T] (* Jean-François Alcover, Sep 27 2021 *)

Formula

E.g.f.: cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1.

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004