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A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1) * x).

%I A348087 #24 May 16 2025 14:34:06
%S A348087 1,1,13,330,12411,618870,38461522,2863440580,248440887123,
%T A348087 24616763946918,2742625188929990,339386813915985836,
%U A348087 46184075261030623710,6854605372617955658940,1101943692701420653738500,190748265085183804327197000,35373318817392757170821576835
%N A348087 a(n) = [x^n] Product_{k=1..n} 1/(1 - (2*k-1) * x).
%H A348087 Seiichi Manyama, <a href="/A348087/b348087.txt">Table of n, a(n) for n = 0..317</a>
%F A348087 a(n) = A039755(2*n-1,n-1) for n > 0.
%F A348087 a(n) = (1/((-2)^(n-1) * (n-1)!)) * Sum_{k=0..n-1} (-1)^k * (2*k+1)^(2*n-1) * binomial(n-1,k) for n > 0.
%F A348087 a(n) ~ 2^(3*n - 1) * n^(n - 1/2) / (sqrt(Pi*(1-c)) * (2-c)^n * c^(n - 1/2) * exp(n)), where c = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599... - _Vaclav Kotesovec_, Oct 02 2021
%F A348087 From _Seiichi Manyama_, May 16 2025: (Start)
%F A348087 a(n) = Sum_{k=0..n} 2^k * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0.
%F A348087 a(n) = Sum_{k=0..n} (-2)^k * (2*n-1)^(n-k) * binomial(2*n-1,k+n-1) * Stirling2(k+n-1,n-1) for n > 0. (End)
%o A348087 (PARI) a(n) = polcoef(1/prod(k=1, n, 1-(2*k-1)*x+x*O(x^n)), n);
%o A348087 (PARI) a(n) = if(n==0, 1, sum(k=0, n-1, (-1)^k*(2*k+1)^(2*n-1)*binomial(n-1, k))/((-2)^(n-1)*(n-1)!));
%Y A348087 Cf. A001147, A007820, A039755, A348085, A348088.
%K A348087 nonn
%O A348087 0,3
%A A348087 _Seiichi Manyama_, Sep 28 2021