A392786 Expansion of e.g.f. (1/x) * Series_Reversion( x - log(1+x^3)/x ).

1, 1, 4, 30, 324, 4620, 82080, 1748040, 43428000, 1233429120, 39429028800, 1401209409600, 54803710684800, 2339492959964160, 108242143312343040, 5395595994409420800, 288278602016461056000, 16434807911438486169600, 995820904914218269286400

Offset

0,3

Comments

a(206) is negative. - Vaclav Kotesovec, Jan 23 2026

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 - log(1+(x*A(x))^3)/(x*A(x))^2).

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

Mathematica

Table[1/(n+1) * Sum[(2*n-3*k)! * StirlingS1[n-2*k, n-3*k] / (n-2*k)!, {k, 0, n/3}], {n, 0, 20}]
(* or *)
nmax = 20; CoefficientList[1/x*InverseSeries[Series[x - Log[1 + x^3]/x, {x, 0, nmax+1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 23 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x-log(1+x^3)/x)/x))

Crossrefs

Cf. A392789.

Sequence in context: A128329 A211828 A277759 * A387974 A006149 A207833

Adjacent sequences: A392783 A392784 A392785 * A392787 A392788 A392789

Keyword

changed,sign

Author

Seiichi Manyama, Jan 22 2026

Status

approved