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A348085 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1) * x).

%I A348085 #24 May 16 2025 14:34:15
%S A348085 1,4,170,13776,1652442,262842580,52116296024,12380577235040,
%T A348085 3427841258566890,1083931844930932140,385417972804020879450,
%U A348085 152219732613102667656000,66113646914860527721527960,31319437721634527178263452656
%N A348085 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1) * x).
%H A348085 Seiichi Manyama, <a href="/A348085/b348085.txt">Table of n, a(n) for n = 0..277</a>
%F A348085 a(n) = A039755(3*n-1,2*n-1) for n > 0.
%F A348085 a(n) = (-1/(2^(2*n-1) * (2*n-1)!)) * Sum_{k=0..2*n-1} (-1)^k * (2*k+1)^(3*n-1) * binomial(2*n-1,k) for n > 0.
%F A348085 a(n) ~ 3^(3*n - 1/2) * n^(n - 1/2) / (sqrt(2*Pi*(1-c)) * (3 - 2*c)^n * c^(2*n - 1/2) * exp(n)), where c = -LambertW(-3*exp(-3/2)/2) = 0.62578253420128292... - _Vaclav Kotesovec_, Oct 02 2021
%F A348085 From _Seiichi Manyama_, May 16 2025: (Start)
%F A348085 a(n) = Sum_{k=0..n} 2^k * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0.
%F A348085 a(n) = Sum_{k=0..n} (-2)^k * (4*n-1)^(n-k) * binomial(3*n-1,k+2*n-1) * Stirling2(k+2*n-1,2*n-1) for n > 0. (End)
%o A348085 (PARI) a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)*x+x*O(x^n)), n);
%o A348085 (PARI) a(n) = if(n==0, 1, -sum(k=0, 2*n-1, (-1)^k*(2*k+1)^(3*n-1)*binomial(2*n-1, k))/(2^(2*n-1)*(2*n-1)!));
%Y A348085 Cf. A039755, A293318, A348082, A348084, A348087.
%K A348085 nonn
%O A348085 0,2
%A A348085 _Seiichi Manyama_, Sep 28 2021