A380043 - OEIS

%I #9 Jan 11 2025 10:27:43

%S 1,1,6,73,1364,34585,1110406,43200535,1975744856,103892750209,

%T 6176282882570,409635957376591,29988473838531748,2402004132488328433,

%U 208956515057627326094,19619264794744128427495,1977503574407863125008816,212975277029523353673126529,24408338689788753822318157330

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

%F E.g.f.: ( (1/x) * Series_Reversion(x/(1 + 3*x*exp(x))) )^(1/3).

%F a(n) = (n!/(3*n+1)) * Sum_{k=0..n} 3^k * k^(n-k) * binomial(n+1/3,k)/(n-k)!.

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x/(1+3*x*exp(x)))/x)^(1/3)))

%o (PARI) a(n) = n!*sum(k=0, n, 3^k*k^(n-k)*binomial(n+1/3, k)/(n-k)!)/(3*n+1);

%Y Cf. A380039, A380041.

%Y Cf. A161633, A380042.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 10 2025