OFFSET
1,5
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 Integrate(cos(n*x)*sin(x)^10,{x,-Pi,Pi}), 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
FORMULA
E.g.f.: cosh(sqrt(y)*arcsin(x))+sqrt(y)*sinh(sqrt(y)*arcsin(x))-1.
EXAMPLE
Triangle starts:
1;
1;
1, 1;
4, 1;
9, 10, 1;
64, 20, 1;
225, 259, 35, 1;
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 *)
CROSSREFS
KEYWORD
nonn,tabf,easy
AUTHOR
Karol A. Penson, Feb 08 2004
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004
STATUS
approved