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 A348081 #22 May 17 2025 03:54:14
%S A348081 1,5,627,251498,209609235,298201326150,646748606934510,
%T A348081 1986821811445598260,8209989926930833199235,
%U A348081 43919039258570117113742270,295300365118450495520630242042,2437724587984574697761809904387340,24239364659088896670563082403144467630
%N A348081 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - k^2 * x).
%H A348081 Seiichi Manyama, <a href="/A348081/b348081.txt">Table of n, a(n) for n = 0..180</a>
%F A348081 a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 78.52705817932973261726305432915417900827309581709564977985583533852704254... = (2+r)^6 / (r^2*(4+r)^2), where r = 0.329482909104375658581668801636329590897344... is the root of the equation 4+r = r*exp(6/(2+r)) and c = (2+r)/(Pi^(3/2)*sqrt(32 - 4*r*(4+r))) = 0.0815842039686253664272939415761688591712635596695065951780203519... - _Vaclav Kotesovec_, Oct 16 2021, updated May 17 2025
%F A348081 From _Seiichi Manyama_, May 13 2025: (Start)
%F A348081 a(n) = A036969(3*n,2*n) = A269945(3*n,2*n).
%F A348081 a(n) = (1/(4*n)!) * Sum_{k=0..4*n} (-1)^k * (2*n-k)^(6*n) * binomial(4*n,k).
%F A348081 a(n) = Sum_{k=0..2*n} (-2*n)^k * binomial(6*n,k) * Stirling2(6*n-k,4*n).
%F A348081 a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(2*n+k,2*n) * Stirling2(4*n-k,2*n). (End)
%t A348081 Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 16 2021 *)
%o A348081 (PARI) a(n) = polcoef(1/prod(k=1, 2*n, 1-k^2*x+x*O(x^n)), n);
%Y A348081 Cf. A036969, A234324, A269945, A298851, A348082.
%K A348081 nonn
%O A348081 0,2
%A A348081 _Seiichi Manyama_, Sep 27 2021