A349524 a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling2(n,k).

1, 1, 6, 65, 1059, 23232, 642859, 21507733, 844701160, 38108248719, 1942394699283, 110401966739110, 6923805346540685, 474957822716470901, 35377953843680999326, 2843665890900123673997, 245340865605247369255751, 22614510471168438300336440, 2217985444621941684970200607

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..348

Formula

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

E.g.f.: exp(-LambertW(2 - 2*exp(x))/2).

a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(2*exp(1))) and c = sqrt(exp(1) * (1 + 2*exp(1)) * log(1 + 1/(2*exp(1))) / (2*Pi))/2 = 0.3428481589262346912499652905097648170872882109000404115070292580887155335...

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

E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^2.

G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x). - Seiichi Manyama, Nov 20 2021

Maple

b:= proc(n, m) option remember; `if`(n=0,
     (2*m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

Mathematica

Table[Sum[(2*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Sqrt[-LambertW[2*(-E^x + 1)]/(2*(E^x - 1))], {x, 0, nmax}], x] * Range[0, nmax]!

Prog

(PARI) a(n) = sum(k=0, n, (2*k+1)^(k-1)*stirling(n, k, 2)); \\ Seiichi Manyama, Nov 20 2021
(PARI) N=20; x='x+O('x^N); Vec(sum(k=0, N, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Nov 20 2021

Crossrefs

Cf. A000110, A008277, A052880, A349504, A349525.

Sequence in context: A360977 A243698 A217899 * A006959 A121017 A239998

Adjacent sequences: A349521 A349522 A349523 * A349525 A349526 A349527

Keyword

nonn

Author

Vaclav Kotesovec, Nov 20 2021

Status

approved