A390646 E.g.f. A(x) satisfies A(x) = 1 + x^2*exp(x)*A(x)^2.

1, 0, 2, 6, 60, 500, 6510, 89082, 1525496, 28679688, 620716410, 14749622030, 388665985092, 11137893075036, 346900608246374, 11631152059778370, 418514282556639600, 16069700995539838352, 656271648283317457266, 28395479216289257066646, 1297884590949912183942140

Offset

0,3

Links

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} k^(n-2*k) * Catalan(k)/(n-2*k)!.

E.g.f.: 2/(1 + sqrt(1 - 4*x^2*exp(x))).

a(n) ~ sqrt(1 + LambertW(1/4)) * n^(n-1) / (2^(n-1) * exp(n) * LambertW(1/4)^n). - Vaclav Kotesovec, Nov 18 2025

Mathematica

Join[{1}, Table[Factorial[n]*Sum[k^(n-2*k)*Binomial[2*k, k]/((k+1)*Factorial[n-2*k]), {k, 0, Floor[n/2]}], {n, 1, 20}]] (* Vincenzo Librandi, Dec 27 2025 *)

Prog

(PARI) a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(2*k, k)/((k+1)*(n-2*k)!));
(Magma) [Factorial(n)*&+[k^(n-2*k)*Binomial(2*k, k)/((k+1)*Factorial((n-2*k))): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Dec 27 2025

Crossrefs

Cf. A358080, A390647.

Cf. A000108, A371319, A387714.

Sequence in context: A056604 A376515 A362702 * A356259 A215720 A376492

Adjacent sequences: A390643 A390644 A390645 * A390647 A390648 A390649

Keyword

nonn

Author

Seiichi Manyama, Nov 13 2025

Status

approved