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A365558
Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(2/3).
6
1, 2, 12, 112, 1432, 23272, 458952, 10644552, 283851272, 8555351112, 287585280392, 10666369505992, 432674936431112, 19054822031194952, 905387807689821832, 46166008179076287432, 2514469578906179506952, 145691888630159515550792
OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (3 - k/n) * binomial(n,k) * a(n-k).
a(n) ~ sqrt(3) * Gamma(1/3) * n^(n + 1/6) / (sqrt(Pi) * 2^(11/6) * exp(n) * log(4/3)^(n + 2/3)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 2*a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
MATHEMATICA
a[n_] := Sum[Product[3*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
PROG
(PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 3*j+2)*stirling(n, k, 2));
CROSSREFS
Sequence in context: A374562 A292187 A124213 * A143134 A214225 A185190
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 09 2023
STATUS
approved