A390106 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x/(1-x)) ).

1, 3, 33, 666, 19917, 794808, 39793653, 2400715152, 169632146457, 13746412333440, 1257095590171209, 128082408166626048, 14389586573367121893, 1767342720783195730944, 235601486675890785949725, 33880917488630933862254592, 5228242537142730043216638513, 861748709856716757704405778432

Offset

0,2

Links

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

Formula

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

E.g.f.: exp(3 * B(x)), where B(x) is the e.g.f. of A376094.

E.g.f.: C(x)^3, where C(x) is the e.g.f. of A361094.

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

a(n) ~ (sqrt(21) - 3) * 2^(n - 1/2) * exp((sqrt(21) - 3 + (sqrt(21) - 5)*n)/2) * n^(n-1) / (21^(1/4) * (5 - sqrt(21))^(n + 1/2)). - Vaclav Kotesovec, Oct 25 2025

Mathematica

Table[3 * n! * Sum[(3*n+3)^(k-1) * Binomial[n-1, n-k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 25 2025 *)

Prog

(PARI) a(n, q=3, r=3, s=3, t=1, u=0) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);
(Magma) [3*Factorial(n) * &+[(3*n+3)^(k-1)*Binomial(n-1, n-k) / Factorial(k) : k in [0..n]] : n in [0..25] ]; // Vincenzo Librandi, Nov 29 2025

Crossrefs

Cf. A052873, A390105.

Cf. A361094, A376094.

Sequence in context: A380880 A336636 A364242 * A376390 A395079 A376393

Adjacent sequences: A390103 A390104 A390105 * A390107 A390108 A390109

Keyword

nonn

Author

Seiichi Manyama, Oct 25 2025

Status

approved