A379879 E.g.f. A(x) satisfies A(x) = exp(-x) + x*A(x)^2.

1, 0, 1, 5, 41, 439, 5869, 94275, 1770705, 38102255, 924580181, 24984120523, 744154938361, 24224671103463, 855748556756157, 32604902612628419, 1332864500919743393, 58192519232324179423, 2702582455278623736997, 133037424985668849756603

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

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

a(n) = -n! * Sum_{k=0..n} (-k-1)^(n-k-1) * binomial(2*k,k)/(n-k)!.

a(n) ~ sqrt(1 + LambertW(-1/4)) * n^(n-1) / (2^(3/2) * (-LambertW(-1/4))^(n+1) * exp(n)). - Vaclav Kotesovec, Jan 23 2025

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(2*exp(-x)/(1+sqrt(1-4*x*exp(-x)))))
(PARI) a(n) = -n!*sum(k=0, n, (-k-1)^(n-k-1)*binomial(2*k, k)/(n-k)!);

Crossrefs

Cf. A000166, A379878.

Cf. A377859, A379868.

Sequence in context: A259609 A323213 A083073 * A115257 A225095 A302100

Adjacent sequences: A379876 A379877 A379878 * A379880 A379882 A379884

Keyword

nonn

Author

Seiichi Manyama, Jan 05 2025

Status

approved