This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348082 #26 May 17 2025 03:26:12
%S A348082 1,10,5082,8187608,27350858986,155829826875450,1352947132455198360,
%T A348082 16634466165612256277904,275064994463136775255491210,
%U A348082 5887721317348514340055453080350,158391364687146632772523433272637642,5231238431447353406197858182627897590880
%N A348082 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1)^2 * x).
%H A348082 Seiichi Manyama, <a href="/A348082/b348082.txt">Table of n, a(n) for n = 0..165</a>
%F A348082 a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 4 * (2+r)^6 / (r^2 * (4+r)^2) = 314.10823271731893046905221731661671603309238326838259911942334135410817..., where r = 0.329482909104375658581668801636329590897344041849... is the root of the equation 4+r = r*exp(6/(2+r)) and c = 1/(2*Pi^(3/2)*sqrt(8/(r*(4 + r)) - 1)) = 0.041829340046147280338756273441751288807538817430199591424694081075... - _Vaclav Kotesovec_, Oct 16 2021, updated May 17 2025
%F A348082 a(n) = A381512(n,4*n-1) = (1/(2^(4*n-2)*(4*n-1)!)) * Sum_{k=0..2*n-1} (-1)^k * (4*n-1-2*k)^(6*n-1) * binomial(4*n-1,k) for n > 0. - _Seiichi Manyama_, May 16 2025
%t A348082 Table[SeriesCoefficient[Product[1/(1 - (2*k-1)^2*x), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 16 2021 *)
%o A348082 (PARI) a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)^2*x+x*O(x^n)), n);
%Y A348082 Cf. A008958, A346543, A348081, A381512.
%K A348082 nonn
%O A348082 0,2
%A A348082 _Seiichi Manyama_, Sep 27 2021