A397280 Expansion of e.g.f. (1/x) * Series_Reversion( x + 1 - exp(x^3) ).

1, 0, 2, 0, 72, 60, 8640, 22680, 2224320, 11975040, 1003363200, 9082735200, 704970604800, 9539017488000, 714339644578560, 13354951401792000, 988680805739827200, 24134163146418124800, 1794899599558281216000, 54820156712956553318400, 4142185546786169186304000

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f. A(x) satisfies A(x) = 1 + (exp((x * A(x))^3) - 1)/x.

a(n) = (1/(n+1)) * Sum_{k=ceiling(n/3)..floor(n/2)} (3*k)!/k! * Stirling2(k,3*k-n).

a(n) ~ sqrt((3*w^2*(1+w) - 1)/(3*w*(2 + 3*w^3))) * n^(n-1) / (exp(n) * (1+w - 1/(3*w^2))^(n+1)), where w = (2*LambertW(1/(2*sqrt(3)))/3)^(1/3). - Vaclav Kotesovec, Jun 20 2026

Mathematica

Table[1/(n+1) * Sum[(3*k)!/k! * StirlingS2[k, 3*k-n], {k, Ceiling[n/3], Floor[n/2]}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 20 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x+1-exp(x^3))/x))

Crossrefs

Cf. A392788, A397281.

Sequence in context: A394970 A306061 A195209 * A397282 A098276 A335692

Adjacent sequences: A397272 A397276 A397277 * A397281 A397282 A397283

Keyword

nonn

Author

Seiichi Manyama, Jun 20 2026

Status

approved