|
|
A009078
|
|
Expansion of e.g.f. cos(tan(x)*sin(x)) (even powers only).
|
|
1
|
|
|
1, 0, -12, -120, -2352, -75840, -1649472, 118634880, 41344643328, 9528901232640, 2213829515240448, 559192086549719040, 156367986602421669888, 48476425507418343751680, 16569615994864645076533248
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
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
|
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|