A355164 a(n) = exp(-1/3) * Sum_{k>=0} (3*k + 2)^n / (3^k * k!).

1, 3, 12, 63, 405, 3024, 25515, 239355, 2465478, 27600669, 333051669, 4303119330, 59202612693, 863285928327, 13288589222508, 215177742933579, 3654114236490393, 64902307993517160, 1202782377224829015, 23207417212751493327, 465302639045308247262, 9677171073270491712513, 208434297638273958963225

Offset

0,2

Links

Michael De Vlieger, Table of n, a(n) for n = 0..484

Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.

Formula

E.g.f.: exp(2*x + (exp(3*x) - 1) / 3).

a(0) = 1; a(n) = 2 * a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 3^(k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A004212(k).

a(n) ~ 3^(n + 2/3) * n^(n + 2/3) * exp(n/LambertW(3*n) - n - 1/3) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^(n + 2/3)). - Vaclav Kotesovec, Jun 27 2022

Mathematica

nmax = 22; CoefficientList[Series[Exp[2 x + (Exp[3 x] - 1)/3], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + Sum[Binomial[n - 1, k - 1] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] 2^(n - k) 3^k BellB[k, 1/3], {k, 0, n}], {n, 0, 22}]

Crossrefs

Cf. A003575, A004212, A284864, A355165, A355167.

Sequence in context: A308206 A264151 A186186 * A238887 A351778 A135889

Adjacent sequences: A355161 A355162 A355163 * A355165 A355166 A355167

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 22 2022

Status

approved