A392720 Expansion of e.g.f. Series_Reversion(x + log(1-x)^3).

1, 0, 6, 36, 570, 8910, 186144, 4433352, 123279024, 3864412800, 135568353096, 5251483423296, 222801862422672, 10271930454467136, 511403120453971584, 27343592521263915840, 1562696309425347849600, 95063362319521889095680, 6133093415308264664044800

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Formula

a(n) = Sum_{k=0..floor((n-1)/2)} (3*k)!/k! * |Stirling1(n-1+k,3*k)|.

E.g.f. A(x) satisfies A(x) = x - log(1 - A(x))^3.

a(n) ~ 2*LambertW(1/(2*sqrt(3)))^(3/2) * sqrt(3/(1 + LambertW(1/(2*sqrt(3))))) * n^(n-1) / (exp(n) * (1 - 4*LambertW(1/(2*sqrt(3)))^2 * (3 + 2*LambertW(1/(2*sqrt(3)))))^(n - 1/2)). - Vaclav Kotesovec, Jan 31 2026

Mathematica

Table[Sum[(3*k)!/k! * Abs[StirlingS1[n-1+k, 3*k]], {k, 0, (n-1)/2}], {n, 1, 20}] (* Vaclav Kotesovec, Jan 31 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x+log(1-x)^3)))
(Magma) [ &+[Factorial(3*k)/Factorial(k)*Abs(StirlingFirst(n-1+k, 3*k)): k in [0..Floor((n-1)/2)]]: n in [1..25]]; // Vincenzo Librandi, Feb 14 2026

Crossrefs

Cf. A143154, A392722.

Cf. A392719.

Sequence in context: A367488 A392716 A339300 * A296389 A077290 A222780

Adjacent sequences: A392717 A392718 A392719 * A392721 A392722 A392723

Keyword

nonn

Author

Seiichi Manyama, Jan 20 2026

Status

approved