A353998 Expansion of e.g.f. 1/(1 - x^2/2 * (exp(x) - 1)).

1, 0, 0, 3, 6, 10, 195, 1281, 5908, 68076, 758565, 6486535, 75598446, 1059484218, 13378016743, 185273328345, 2999003869800, 48665352612376, 816394913567433, 15110162148144267, 292156921946387170, 5805684093139498470, 122617308231635240331

Offset

0,4

Links

Table of n, a(n) for n=0..22.

Formula

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

a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

a(n) ~ 2 * n! / ((4 + 2*r + r^3) * r^n), where r = 1.043121496712693605897520269472163423276582653660720448... is the root of the equation (exp(r)-1)*r^2 = 2. - Vaclav Kotesovec, May 13 2022

Prog

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

Crossrefs

Cf. A052848, A353999.

Cf. A351505, A354000.

Sequence in context: A254957 A124266 A137941 * A355181 A356951 A355179

Adjacent sequences: A353995 A353996 A353997 * A353999 A354000 A354001

Keyword

nonn

Author

Seiichi Manyama, May 13 2022

Status

approved