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%I #24 Nov 17 2023 11:20:24
%S 1,4,44,764,18204,551644,20291804,877970524,43680345564,2456429581404,
%T 154072160204764,10663000409493084,807124301044917724,
%U 66329628496719183964,5881222650127663682524,559616682597652939940444,56879286407092006924382684
%N Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(2/3).
%F a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+2)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+4)) * Stirling2(n,k).
%F a(0) = 1; a(n) = Sum_{k=1..n} (6 - 2*k/n) * binomial(n,k) * a(n-k).
%F O.g.f. (conjectural): 1/(1 - 4*x/(1 - 7*x/(1 - 10*x/(1 - 14*x/(1 - 16*x/(1 - 21*x/(1 - ... - (6*n - 2)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - _Peter Bala_, Sep 24 2023
%F a(n) ~ Gamma(1/3) * sqrt(3) * n^(n + 1/6) / (sqrt(2*Pi) * 7^(2/3) * exp(n) * log(7/6)^(n + 2/3)). - _Vaclav Kotesovec_, Nov 11 2023
%F a(0) = 1; a(n) = 4*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - _Seiichi Manyama_, Nov 17 2023
%t a[n_] := Sum[Product[6*j + 4, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* _Amiram Eldar_, Sep 11 2023 *)
%o (PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+4)*stirling(n, k, 2));
%Y Cf. A094419, A346985, A354252, A365555, A365557.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Sep 09 2023
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