A373324 E.g.f. satisfies A(x) = exp(x) + x*A(x)^4.

1, 2, 17, 349, 11249, 495401, 27715225, 1882157369, 150415131809, 13830661215649, 1438437863857961, 166962406866895817, 21396540301232809201, 3000661115664455591921, 457109095827413086174265, 75165845570197217863619161, 13270031366484750565975875905, 2503433069466253671859276038977

Offset: 0

Author

Vaclav Kotesovec, Jun 01 2024

Keywords

• nonn

Comments

In general, for k > 1, if e.g.f. satisfies A(x) = exp(x) + x*A(x)^k, then a(n) ~ sqrt(1 + LambertW((1 - 1/k)^k)) * (k-1)^(n - 1/2 + 1/(k-1)) * n^(n-1) / (k^(1/2 + 1/(k-1)) * exp(n) * LambertW((1 - 1/k)^k)^(n + 1/(k-1))).

Crossrefs

Cf. A000522, A194471, A371318.

Programs

Mathematica

Table[n! * Sum[(3*k+1)^(n-k-1) * Binomial[4*k,k] / (n-k)!, {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ sqrt(1 + LambertW(81/256)) * 3^(n - 1/6) * n^(n-1) / (2^(5/3) * exp(n) * LambertW(81/256)^(n + 1/3)).