A365567 Expansion of e.g.f. 1 / (5 - 4 * exp(x))^(3/4).
1, 3, 24, 297, 5001, 106578, 2748399, 83182347, 2890153626, 113364686403, 4954547485149, 238734066994272, 12573018414279501, 718498413957515103, 44278797576715884024, 2927171415480872824197, 206625238881832412874501, 15511299587628626891270178
Offset: 0
Keywords
Programs
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Mathematica
a[n_] := Sum[Product[4*j + 3, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *) -
PARI
a(n) = sum(k=0, n, prod(j=0, k-1, 4*j+3)*stirling(n, k, 2));
Formula
a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (4*j+3)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (4 - k/n) * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/4) * n^(n + 1/4) / (5^(3/4) * sqrt(Pi) * exp(n) * log(5/4)^(n + 3/4)). - Vaclav Kotesovec, Nov 11 2023
a(0) = 1; a(n) = 3*a(n-1) - 5*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023