A391555 Expansion of e.g.f. exp((g^4 - 1)/4), where g = 1+x*g^2 is the g.f. of A000108.

1, 1, 8, 94, 1468, 28676, 673096, 18456208, 578989664, 20451955168, 803281220416, 34725507491456, 1638442250105728, 83783085867535744, 4615534061313644288, 272513023397196560896, 17167327855852316678656, 1149361306938635404179968, 81495563955795289474269184

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(2*n+4*k+4,n)/((2*n+4*k+4) * 4^k * k!) for n > 0.

From Vaclav Kotesovec, Dec 21 2025: (Start)

Recurrence: (n^2 + 11*n - 70)*a(n) = (7*n^3 + 76*n^2 - 556*n + 415)*a(n-1) - (n-1)*(13*n^3 + 137*n^2 - 1161*n + 1685)*a(n-2) + (n-2)*(n-1)*(4*n^3 + 42*n^2 - 347*n + 545)*a(n-3) - (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(n^2 + 16*n + 10)*a(n-4) + 2*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(2*n - 5)*(n^2 + 13*n - 58)*a(n-5).

a(n) ~ 2^(2*n + 7/2) * n^(n-1) / exp(n - 15/4). (End)

Mathematica

nmax=20; g[x_]:=Sum[Binomial[2*k, k]/(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(2*k, k)/(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(2*k, k)/(k+1)*x^k:k in [0..N]])^4-1)/4), n):n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A000108, A391544.

Sequence in context: A386895 A098269 A010565 * A299002 A299669 A080208

Adjacent sequences: A391552 A391553 A391554 * A391556 A391557 A391558

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved