A009050 Expansion of e.g.f. cos(x*sin(x)) (even power only).

1, 0, -12, 120, 784, -95040, 3292608, -9423232, -9230042880, 890079012864, -40083887897600, -2468650419873792, 837675174905843712, -109581967679961088000, 5834566862721760149504

Offset

0,3

Links

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

Formula

a(n) = 2*Sum_{k=1..n-1} binomial(2*n,2*k)*(4^(n-2*k)*(-1)^(k)* Sum_{i=0..k-1} (i-k)^(2*n-2*k)*binomial(2*k,i)*(-1)^(n-i)), n>0, a(0)=1. - Vladimir Kruchinin, Jun 30 2011

Maple

seq(coeff(series(factorial(n)*cos(x*sin(x)), x, n+1), x, n), n=0..40, 2); # Muniru A Asiru, Jul 24 2018

Mathematica

With[{nmax = 60}, CoefficientList[Series[Cos[x*Sin[x]], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* G. C. Greubel, Jul 23 2018 *)

Prog

(Maxima)
a(n):=if n=0 then 1 else 2*sum(binomial(2*n, 2*k)*(4^(n-2*k)*(-1)^(k)*sum((i-k)^(2*n-2*k)*binomial(2*k, i)*(-1)^(n-i), i, 0, k-1)), k, 1, n-1); /* Vladimir Kruchinin, Jun 30 2011 */
(PARI) x='x+O('x^60); v=Vec(serlaplace(cos(x*sin(x)))); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, Jul 23 2018
(GAP) Concatenation([1], List([1..15], n->2*Sum([1..n-1], k->Binomial(2*n, 2*k)*(4^(n-2*k)*(-1)^k)*Sum([0..k-1], i->(i-k)^(2*n-2*k)*Binomial(2*k, i)*(-1)^(n-i))))); # Muniru A Asiru, Jul 24 2018

Crossrefs

Sequence in context: A305624 A056320 A056311 * A067358 A268634 A061506

Adjacent sequences: A009047 A009048 A009049 * A009051 A009052 A009053

Keyword

sign

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Status

approved