A391545 Expansion of e.g.f. exp(g^2 - 1), where g = 1+x*g^3 is the g.f. of A001764.

1, 2, 18, 272, 5812, 161112, 5500744, 223415936, 10529259024, 565135093664, 34045145461024, 2275337355049728, 167102361753877312, 13378718058724535168, 1159904199850601405568, 108268982677754935390208, 10826634986211198785741056, 1154770136003332797563318784

Offset

0,2

Links

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A391556.

a(n) = 2 * n! * exp(-1) * Sum_{k>=0} binomial(3*n+2*k+2,n)/((3*n+2*k+2) * k!) for n > 0.

Mathematica

Table[Factorial[n] SeriesCoefficient[Exp[(Sum[Binomial[3 k, k]/(2*k+1) x^k, {k, 0, 20}])^2-1], {x, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Dec 21 2025 *)

Prog

(PARI) my(N=20, x='x+O('x^N), g=sum(k=0, N, binomial(3*k, k)/(2*k+1)*x^k)); Vec(serlaplace(exp(g^2-1)))
(Magma) N := 20; R<x> := PowerSeriesRing(Rationals(), 2*N); [Coefficient(Exp((&+[Binomial(3*k, k)/(2*k+1) * x^k : k in [0..n]])^2-1), n)*Factorial(n): n in [0..N]]; // Vincenzo Librandi, Dec 21 2025

Crossrefs

Cf. A391546, A391547.

Cf. A001764, A391556.

Sequence in context: A351275 A364400 A377503 * A334242 A349314 A365555

Adjacent sequences: A391542 A391543 A391544 * A391546 A391547 A391548

Keyword

nonn

Author

Seiichi Manyama, Dec 13 2025

Status

approved