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A355164
a(n) = exp(-1/3) * Sum_{k>=0} (3*k + 2)^n / (3^k * k!).
4
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 22 2022
STATUS
approved