A392785 - OEIS

%I #18 Jan 25 2026 04:02:53

%S 1,1,4,30,336,5100,97920,2275560,62092800,1946064960,68894582400,

%T 2719175659200,118394186803200,5637541432723200,291441967584768000,

%U 16256535509069856000,973201947936970752000,62239242631037010739200,4234925802836604395520000

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

%H Vincenzo Librandi, <a href="/A392785/b392785.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f. A(x) satisfies A(x) = 1/(1 + log(1-(x*A(x))^4)/(x*A(x))^3).

%F 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)!.

%t 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 *)

%t 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 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x+log(1-x^4)/x^2)/x))

%o (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

%Y Cf. A052802, A213641, A392784.

%Y Cf. A392787, A392790.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 22 2026