A349525 a(n) = Sum_{k=0..n} (3*k+1)^(k-1) * Stirling2(n,k).

1, 1, 8, 122, 2847, 90112, 3611162, 175352515, 10009442658, 656934750150, 48744407335597, 4035143806865514, 368706775967717518, 36861117438297883213, 4002400525694764367212, 469049713401827161071110, 59010099414303871987517111, 7932542361585921797125908876

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..331

Formula

E.g.f.: (-LambertW(3*(-exp(x) + 1)) / (3*(exp(x) - 1)))^(1/3).

E.g.f.: exp(-LambertW(3 - 3*exp(x))/3).

a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(3*exp(1))) and c = exp(1/3) * sqrt((1 + 3*exp(1)) * log(1 + 1/(3*exp(1))) / (2*Pi))/3 = 0.190981550465823640438134269765128596177617920807463710992027181154754728...

a(n) ~ sqrt(1 + 3*exp(1)) * n^(n-1) / (3*exp(n - 1/3) * log(1 + 1/(3*exp(1)))^(n - 1/2)).

E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^3.

G.f.: Sum_{k>=0} (3*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,
     (3*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[(3*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[(-LambertW[3*(-E^x + 1)]/(3*(E^x - 1)))^(1/3), {x, 0, nmax}], x] * Range[0, nmax]!

Prog

(PARI) a(n) = sum(k=0, n, (3*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, (3*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Nov 20 2021

Crossrefs

Cf. A000110, A008277, A052880, A349505, A349524.

Sequence in context: A222498 A361308 A359355 * A240477 A195247 A239755

Adjacent sequences: A349522 A349523 A349524 * A349526 A349527 A349528

Keyword

nonn

Author

Vaclav Kotesovec, Nov 20 2021

Status

approved