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
KEYWORD
sign
AUTHOR
Vladimir Kruchinin, Apr 21 2011
STATUS
approved