A009078 Expansion of e.g.f. cos(tan(x)*sin(x)) (even powers only).

1, 0, -12, -120, -2352, -75840, -1649472, 118634880, 41344643328, 9528901232640, 2213829515240448, 559192086549719040, 156367986602421669888, 48476425507418343751680, 16569615994864645076533248

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..240

Formula

a(n) = Sum_{k=1..n} (4^(n-k)*Sum_{t=k..n-k} binomial(2*n,2*t)*((Sum_{j=2*k..2*n-2*t} binomial(j-1,2*k-1)*j!*stirling2(2*n-2*t,j)*(-1)^(n+j)*2^(1-j)))*sum(i=0..k, (i-k)^(2*t)*binomial(2*k,i)*(-1)^(k-i)))/(2*k)!, n>0, a(0)=1. - Vladimir Kruchinin, Jun 30 2011

Mathematica

With[{nn=30}, Take[CoefficientList[Series[Cos[Tan[x]Sin[x]], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, Jun 04 2018 *)

Prog

(Maxima)
a(n):=if n=0 then 1 else sum((4^(n-k)*sum(binomial(2*n, 2*t)*((sum(binomial(j-1, 2*k-1)*j!*stirling2(2*n-2*t, j)*(-1)^(n+j)*2^(1-j), j, 2*k, 2*n-2*t))*sum((i-k)^(2*t)*binomial(2*k, i)*(-1)^(k-i), i, 0, k)), t, k, n-k))/(2*k)!, k, 1, n); /* Vladimir Kruchinin, Jun 30 2011 */
(PARI) x='x+O('x^50); v=Vec(serlaplace(cos(tan(x)*sin(x)))); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, Jul 24 2018

Crossrefs

Sequence in context: A012621 A102341 A174561 * A221493 A009149 A010570

Adjacent sequences: A009075 A009076 A009077 * A009079 A009080 A009081

Keyword

sign

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Status

approved