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%I #17 Nov 18 2023 08:27:20
%S 1,1,9,121,2289,56401,1713849,61939081,2595199329,123690992161,
%T 6608289658089,391154820258841,25408740616159569,1797051730819428721,
%U 137463201511019813529,11308020549364112399401,995455518982520306979009,93373681491447943767190081
%N Expansion of e.g.f. 1/(2 - exp(4*x))^(1/4).
%F a(n) = Sum_{k=0..n} 4^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
%F a(n) ~ n! * 2^(2*n - 1/4) / (Gamma(1/4) * n^(3/4) * log(2)^(n + 1/4)). - _Vaclav Kotesovec_, Mar 05 2022
%F From _Seiichi Manyama_, Nov 18 2023: (Start)
%F a(0) = 1; a(n) = Sum_{k=1..n} 4^k * (1 - 3/4 * k/n) * binomial(n,k) * a(n-k).
%F a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-4)^k * binomial(n-1,k) * a(n-k). (End)
%t m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[4*x])^(-1/4), {x, 0, m}], x] (* _Amiram Eldar_, Mar 05 2022 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(4*x))^(1/4)))
%o (PARI) a(n) = sum(k=0, n, 4^(n-k)*prod(j=0, k-1, 4*j+1)*stirling(n, k, 2));
%Y Cf. A000670, A352117, A352118.
%Y Cf. A352073.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Mar 05 2022
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