A349559 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^x.

1, 0, 2, 3, 44, 270, 3714, 44940, 746528, 13149864, 271954440, 6154715160, 155055594792, 4254730262640, 127019898548256, 4088313657038520, 141301521555548160, 5216698433745207360, 204946906542573645504, 8536144551987171202560

Offset

0,3

Links

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

Formula

a(n) ~ sqrt(1 + r - 2*r*log(r)) * n^(n-1) / ((1+r)^2 * exp(n) * r^(n + 1/2)), where r = 0.4214518303433019663622598075106479936652984008256... is the root of the equation r^(1-r) * (1+r)^(1+r) = 1. - Vaclav Kotesovec, Nov 22 2021

a(n) = n! * Sum_{k=0..floor(n/2)} (n-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 27 2022

Maple

a:= n-> n!*coeff(series(RootOf(1/(1-x*A)^x-A, A), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 22 2021

Mathematica

nmax = 20; A[_] = 0; Do[A[x_] = 1/(1 - x*A[x])^x + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 22 2021 *)

Prog

(PARI) my(A=1, n=22); for(i=1, n, A=(1-x*A)^(-x+x*O(x^n))); Vec(serlaplace(A))
(PARI) a(n) = n!*sum(k=0, n\2, (n-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, Aug 27 2022

Crossrefs

Cf. A007113, A349556.

Cf. A184949, A356786, A356787.

Sequence in context: A100443 A334243 A323619 * A349827 A060415 A289661

Adjacent sequences: A349556 A349557 A349558 * A349560 A349561 A349562

Keyword

nonn

Author

Seiichi Manyama, Nov 22 2021

Status

approved