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A365777 Expansion of e.g.f. (exp(2*x) / (2 - exp(2*x)))^(3/4). 2

%I #13 Nov 16 2023 11:51:46

%S 1,3,15,117,1257,17163,284055,5522877,123344817,3111071283,

%T 87454712895,2710961144037,91862770847577,3378032307195003,

%U 133970268354806535,5699864583381903597,258956671286986317537,12512342291081486212323,640686944845321006836975

%N Expansion of e.g.f. (exp(2*x) / (2 - exp(2*x)))^(3/4).

%F a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (4*j+3)) * Stirling2(n,k).

%F a(0) = 1; a(n) = Sum_{k=1..n} (-2)^k * (1/2 * k/n - 2) * binomial(n,k) * a(n-k).

%F a(0) = 1; a(n) = 3*a(n-1) + Sum_{k=1..n-1} 2^k * binomial(n-1,k) * a(n-k).

%o (PARI) a(n) = sum(k=0, n, (-2)^(n-k)*prod(j=0, k-1, 4*j+3)*stirling(n, k, 2));

%Y Cf. A124212, A276371.

%Y Cf. A365794.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Nov 16 2023

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