A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).

1, 1, 4, 36, 484, 8840, 203868, 5691308, 186612592, 7031373264, 299397454080, 14218443479328, 745142534904480, 42717896158340832, 2659373970144454080, 178666030775042040000, 12884568940594969258752, 992750028716940749121792

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..356

Formula

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling1(n,k).

E.g.f. A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (log(1+x))^k / k!.

From Vaclav Kotesovec, Nov 20 2021: (Start)

E.g.f.: sqrt(-LambertW(-2*log(1 + x)) / (2*log(1 + x))).

a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1)). (End)

Mathematica

nmax = 20; CoefficientList[Series[Sqrt[-LambertW[-2*Log[1 + x]]/(2*Log[1 + x])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)

Prog

(PARI) a(n) = sum(k=0, n, (2*k+1)^(k-1)*stirling(n, k, 1));
(PARI) N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*log(1+x)^k/k!)))

Crossrefs

Cf. A008275, A033917, A264407, A349505, A349524.

Sequence in context: A002690 A370927 A094417 * A354264 A138435 A366337

Adjacent sequences: A349501 A349502 A349503 * A349505 A349506 A349507

Keyword

nonn

Author

Seiichi Manyama, Nov 20 2021

Status

approved