A365600 Expansion of e.g.f. 1 / (1 - 4 * log(1 + x))^(3/4).
1, 3, 18, 174, 2292, 38292, 774624, 18399840, 501868416, 15456483840, 530462128896, 20073406663296, 830293158570624, 37267057695192192, 1803930663341528064, 93672204405378891264, 5193925606670524254720, 306280622206497897745920
Offset: 0
Keywords
Programs
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Mathematica
a[n_] := Sum[Product[4*j + 3, {j, 0, k - 1}] * StirlingS1[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 13 2023 *) -
PARI
a(n) = sum(k=0, n, prod(j=0, k-1, 4*j+3)*stirling(n, k, 1));
Formula
a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (4*j+3)) * Stirling1(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (4 - k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/4) * n^(n + 1/4) / (2^(3/2) * sqrt(Pi) * (exp(1/4) - 1)^(n + 3/4) * exp(n - 3/16)). - Vaclav Kotesovec, Nov 10 2023