A355605 Expansion of e.g.f. (1 + x)^(x^2/2).

1, 0, 0, 3, -6, 20, 0, -126, 1260, -4320, 5040, 180180, -2601720, 31309200, -372756384, 4877195400, -70178799600, 1099333347840, -18429818232960, 327676010785200, -6146676161388000, 121301442091851840, -2512746856371628800, 54527094987619716000

Offset

0,4

Links

Table of n, a(n) for n=0..23.

Formula

a(0) = 1; a(n) = -(n-1)!/2 * Sum_{k=3..n} (-1)^k * k/(k-2) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/3)} Stirling1(n-2*k,k)/(2^k * (n-2*k)!).

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^(x^2/2)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*log(1+x))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!/2*sum(j=3, i, (-1)^j*j/(j-2)*v[i-j+1]/(i-j)!)); v;
(PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 1)/(2^k*(n-2*k)!));

Crossrefs

Cf. A007121, A355603.

Sequence in context: A326317 A306522 A290784 * A356912 A176993 A359963

Adjacent sequences: A355602 A355603 A355604 * A355606 A355607 A355608

Keyword

sign

Author

Seiichi Manyama, Jul 09 2022

Status

approved