A189421 Expansion of e.g.f. exp(sin(x)+2*sin(x)^2).

1, 1, 5, 12, 53, 152, 361, -168, -16055, -123200, -779827, -2504832, 18694397, 338660480, 3543246193, 19320001536, -64409565935, -2591458500608, -36445173712747, -254934852857856, 809555224961861, 46263863308992512, 744209131179612025, 5617003290092961792

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..65

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

a(n) = sum(m=1..n, sum(k=m..n, (binomial(m,k-m)*((-1)^(n-k)+1)*sum(i=0..k/2, (2*i-k)^n*binomial(k,i)*(-1)^((n+k)/2-i))))/(2^m*m!)), n>0, a(0)=1.

Mathematica

a[n_] := Sum[Sum[Binomial[m, k-m] ((-1)^(n-k)+1) Sum[(2i-k)^n Binomial[k, i] (-1)^((n+k)/2-i), {i, 0, k/2}], {k, m, n}]/(2^m m!), {m, 1, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Aug 08 2018, from Maxima *)
Range[0, 23]! CoefficientList[ Series[ Exp[Sin[x] + 2 Sin[x]^2], {x, 0, 23}], x] (* Robert G. Wilson v, Aug 08 2018 *)

Prog

(Maxima)
a(n):=sum(sum((binomial(m, k-m)*((-1)^(n-k)+1)*sum((2*i-k)^n*binomial(k, i)*(-1)^((n+k)/2-i), i, 0, k/2)), k, m, n)/(2^m*m!), m, 1, n);
(PARI)
x='x+O('x^66); /* that many terms */
Vec(serlaplace(exp(sin(x)+2*sin(x)^2))) /* show terms */ /* Joerg Arndt, Apr 25 2011 */

Crossrefs

Sequence in context: A054661 A188118 A062978 * A066961 A179994 A131549

Adjacent sequences: A189418 A189419 A189420 * A189422 A189423 A189424

Keyword

sign

Author

Vladimir Kruchinin, Apr 21 2011

Status

approved