A009015 Expansion of e.g.f.: cos(x*cos(x)) (even powers only).

1, -1, 13, -181, 3865, -140521, 6324517, -344747677, 23853473329, -1996865965009, 193406280000061, -21615227339380357, 2778071540350106953, -403985610499148666041, 65635628800688339178325, -11851572489741709802698861, 2366329597668159364674862177

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Programs

Maple

seq(coeff(series(factorial(n)*(cos(x*cos(x))), x,n+1),x,n),n=0..30,2); # Muniru A Asiru, Jul 21 2018

Mathematica

With[{nmax = 60}, CoefficientList[Series[Cos[x*Cos[x]], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* G. C. Greubel, Jul 21 2018 *)

Maxima

a(n):=sum(binomial(2*n,2*j)*((sum((j-i)^(2*n-2*j)*binomial(2*j,i),i,0,((2*j-1)/2)))*(-1)^(n))/(2^(4*j-2*n-1)),j,0,(2*n-1)/2)+(-1)^n; /* Vladimir Kruchinin, Jun 06 2011 */

PARI

my(x='x+O('x^50)); v=Vec(serlaplace(cos(x*cos(x)))); vector(#v\2,n,v[2*n-1]) \\ G. C. Greubel, Jul 21 2018

Formula

a(n) = (-1)^n + (-1)^n*Sum_{j=0..(2*n-1)/2} binomial(2*n,2*j)*(Sum_{i=0..(2*j-1)/2} (j-i)^(2*n-2*j)*binomial(2*j,i))/(2^(4*j-2*n-1)). - Vladimir Kruchinin, Jun 06 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997