A392790 Expansion of e.g.f. (1/x) * Series_Reversion( x + (1 - exp(x^4))/x^2 ).

1, 1, 4, 30, 336, 5100, 97920, 2275560, 62092800, 1946004480, 68887324800, 2718570254400, 118349479987200, 5634364354819200, 291216292048281600, 16240198896649728000, 971984105567354880000, 62145219629409857740800, 4227385588071740430336000

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 + (1 - exp((x*A(x))^4))/(x*A(x))^3).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (2*n-4*k)! * Stirling2(n-3*k,n-4*k)/(n-3*k)!.

Mathematica

Table[(1/(n+1))*Sum[(2*n-4*k)!*Abs[StirlingS2[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+(1-exp(x^4))/x^2)/x))
(Magma) [ 1/(n+1)* &+[ Factorial(2*n-4*k)*StirlingSecond(n-3*k, n-4*k)/Factorial(n-3*k) : k in [0..Floor(n/4)] ] : n in [0..18] ]; // Vincenzo Librandi, Jan 24 2026

Crossrefs

Cf. A052894, A392788, A392789.

Sequence in context: A392787 A001761 A292220 * A392785 A099712 A370931

Adjacent sequences: A392787 A392788 A392789 * A392791 A392792 A392793

Keyword

nonn

Author

Seiichi Manyama, Jan 22 2026

Status

approved