A391561 Expansion of e.g.f. exp((g^4 - 1)/4), where g = 1+x*g^4 is the g.f. of A002293.

1, 1, 12, 244, 7084, 268716, 12618616, 707542312, 46164369504, 3437762234464, 287830529920576, 26772093907478976, 2739431829975444352, 305869238654653788544, 37010335831481152199424, 4824682223713815167769856, 674162053956687816905529856, 100526605452826479060450949632

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

a(n) = n! * exp(-1/4) * Sum_{k>=0} binomial(4*n+4*k+4,n)/((4*n+4*k+4) * 4^k * k!) for n > 0.

Mathematica

nmax=20; g[x_]:=Sum[Binomial[4*k, k]/(3*k+1) x^k, {k, 0, nmax}];
Table[n! SeriesCoefficient[Exp[(g[x]^4-1)/4], {x, 0, n}], {n, 0, nmax}] (* Vincenzo Librandi, Dec 22 2025 *)

Prog

(PARI) my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); Vec(serlaplace(exp((g^4-1)/4)))
(Magma) N:=20; R<x>:=PowerSeriesRing(Rationals(), 2*N+1); [Factorial(n)*Coefficient(Exp(((&+[Binomial(4*k, k)/(3*k+1)*x^k:k in [0..N]])^4-1)/4), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A002293, A391550.

Sequence in context: A220216 A126687 A064749 * A009472 A012066 A386867

Adjacent sequences: A391558 A391559 A391560 * A391562 A391563 A391564

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved