A348081 a(n) = [x^n] Product_{k=1..2*n} 1/(1 - k^2 * x).
1, 5, 627, 251498, 209609235, 298201326150, 646748606934510, 1986821811445598260, 8209989926930833199235, 43919039258570117113742270, 295300365118450495520630242042, 2437724587984574697761809904387340, 24239364659088896670563082403144467630
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..180
Programs
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Mathematica
Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, 2*n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 16 2021 *) -
PARI
a(n) = polcoef(1/prod(k=1, 2*n, 1-k^2*x+x*O(x^n)), n);
Formula
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
From Seiichi Manyama, May 13 2025: (Start)
a(n) = (1/(4*n)!) * Sum_{k=0..4*n} (-1)^k * (2*n-k)^(6*n) * binomial(4*n,k).
a(n) = Sum_{k=0..2*n} (-2*n)^k * binomial(6*n,k) * Stirling2(6*n-k,4*n).
a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(2*n+k,2*n) * Stirling2(4*n-k,2*n). (End)