A392785 Expansion of e.g.f. (1/x) * Series_Reversion( x + log(1-x^4)/x^2 ).

1, 1, 4, 30, 336, 5100, 97920, 2275560, 62092800, 1946064960, 68894582400, 2719175659200, 118394186803200, 5637541432723200, 291441967584768000, 16256535509069856000, 973201947936970752000, 62239242631037010739200, 4234925802836604395520000

Offset

0,3

Links

Vincenzo Librandi, 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))^4)/(x*A(x))^3).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (2*n-4*k)! * |Stirling1(n-3*k,n-4*k)|/(n-3*k)!.

Mathematica

nmax = 20; CoefficientList[1/x*InverseSeries[Series[x + Log[1 - x^4]/x^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jan 23 2026 *)
Table[(1/(n+1))*Sum[(2*n-4*k)!*Abs[StirlingS1[n-3*k, n-4*k]/(n-3*k)!], {k, 0, Floor[n/4]}], {n, 0, 21}] (* Vincenzo Librandi, Jan 24 2026 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x+log(1-x^4)/x^2)/x))
(Magma) N := 20; R<x> := PowerSeriesRing(Rationals(), 30); f := x + Log(1 - x^4)/x^2; g := Reversion(f); h := g / x; L := R!0; for n in [0..N-1] do L +:= Coefficient(h, n) * Factorial(n) * x^n; end for; [ Coefficient(L, n) : n in [0..N-1] ]; // Vincenzo Librandi, Jan 24 2026

Crossrefs

Cf. A052802, A213641, A392784.

Cf. A392787, A392790.

Sequence in context: A001761 A292220 A392790 * A099712 A370931 A209440

Adjacent sequences: A392782 A392783 A392784 * A392786 A392787 A392788

Keyword

nonn

Author

Seiichi Manyama, Jan 22 2026

Status

approved