A392790 - OEIS

%I #20 Jan 24 2026 09:38:04

%S 1,1,4,30,336,5100,97920,2275560,62092800,1946004480,68887324800,

%T 2718570254400,118349479987200,5634364354819200,291216292048281600,

%U 16240198896649728000,971984105567354880000,62145219629409857740800,4227385588071740430336000

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

%H Vincenzo Librandi, <a href="/A392790/b392790.txt">Table of n, a(n) for n = 0..300</a>

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

%F 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)!.

%t 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 *)

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

%o (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

%Y Cf. A052894, A392788, A392789.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 22 2026