%I #18 Nov 16 2023 11:51:10
%S 1,3,27,387,7659,193491,5948091,215446563,8984708235,423944899443,
%T 22328393101659,1298429924941251,82625791930962219,
%U 5711012035686681363,426058604580805219323,34121803137713388036963,2919847869159667841599947,265868538017899566748612275
%N Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(3/5).
%F a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+3)) * Stirling2(n,k).
%F a(0) = 1; a(n) = Sum_{k=1..n} (5 - 2*k/n) * binomial(n,k) * a(n-k).
%F a(n) ~ sqrt(2*Pi) * n^(n + 1/10) / (6^(3/5) * Gamma(3/5) * exp(n) * log(6/5)^(n + 3/5)). - _Vaclav Kotesovec_, Nov 11 2023
%F a(0) = 1; a(n) = 3*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 16 2023
%t a[n_] := Sum[Product[5*j + 3, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Sep 11 2023 *)
%o (PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+3)*stirling(n, k, 2));
%Y Cf. A094418, A346984, A365568, A365570.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Sep 09 2023
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