A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6*k - 1)^n / (6^k * k!).

1, 0, 6, 36, 324, 3456, 43416, 618192, 9778320, 169827840, 3210376032, 65540155968, 1435094563392, 33510354739200, 830486180748672, 21756166766173440, 600339119317643520, 17394883290643709952, 527782830161632077312, 16727350847049194775552

Offset: 0

Author

Ilya Gutkovskiy, Aug 12 2020

Keywords

• nonn

Comments

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 28 2022

Crossrefs

Cf. A000296, A003578, A005012, A337038, A337039, A337040, A337041, A337043.

Programs

Mathematica

nmax = 19; CoefficientList[Series[Exp[(Exp[6 x] - 1)/6 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 6^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 6^k BellB[k, 1/6], {k, 0, n}], {n, 0, 19}]

Formula

G.f. A(x) satisfies: A(x) = (1 - 6*x + x*A(x/(1 - 6*x))) / (1 - 5*x - 6*x^2).

G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 6*j*x/(1 + x)).

E.g.f.: exp((exp(6*x) - 1) / 6 - x).

a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 6^k * a(n-k-1).

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005012(k).

a(n) ~ 6^(n - 1/6) * n^(n - 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n - 1/6)). - Vaclav Kotesovec, Jun 26 2022