A397280 - OEIS

%I #14 Jun 20 2026 09:02:37

%S 1,0,2,0,72,60,8640,22680,2224320,11975040,1003363200,9082735200,

%T 704970604800,9539017488000,714339644578560,13354951401792000,

%U 988680805739827200,24134163146418124800,1794899599558281216000,54820156712956553318400,4142185546786169186304000

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

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

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

%F 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

%t 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 *)

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

%Y Cf. A392788, A397281.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jun 20 2026