A391550 Expansion of e.g.f. exp(g^4 - 1), where g = 1+x*g^4 is the g.f. of A002293.

1, 4, 60, 1432, 46984, 1964544, 99929056, 5991976960, 414015246528, 32400442245376, 2833433743081216, 273895764519842304, 29006040307478648320, 3340227044053214851072, 415614838629786639120384, 55572930926734714107142144, 7947546303663092651291938816, 1210581403269542242905517670400

Offset

0,2

Links

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

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A391561.

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

Mathematica

Table[Factorial[n] SeriesCoefficient[Exp[(Sum[Binomial[4*k, k]/(3*k+1) x^k, {k, 0, 20}])^4-1], {x, 0, n}], {n, 0, 20}] (* 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)))
(Magma) N := 20; R<x> := PowerSeriesRing(Rationals(), 3*N); [Coefficient(Exp((&+[Binomial(4*k, k)/(3*k+1) * x^k : k in [0..n]])^4-1), n)*Factorial(n): n in [0..N]]; // Vincenzo Librandi, Dec 22 2025

Crossrefs

Cf. A380516, A391548, A391549.

Cf. A002293, A391561.

Sequence in context: A013483 A013484 A013485 * A377528 A324959 A098630

Adjacent sequences: A391547 A391548 A391549 * A391551 A391552 A391553

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved