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a(n) = [x^n] Product_{k=1..2*n} 1/(1 - (2*k-1)^2 * x).
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%I #12 Oct 16 2021 11:04:33

%S 1,10,5082,8187608,27350858986,155829826875450,1352947132455198360,

%T 16634466165612256277904,275064994463136775255491210,

%U 5887721317348514340055453080350,158391364687146632772523433272637642,5231238431447353406197858182627897590880

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

%F a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 314.10823271731893046905221731661671603309238326838259911942334135410817... and c = 0.041829340046147280338756273441751288807538817430199591424694081075... - _Vaclav Kotesovec_, Oct 16 2021

%t 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 (PARI) a(n) = polcoef(1/prod(k=1, 2*n, 1-(2*k-1)^2*x+x*O(x^n)), n);

%Y Cf. A008958, A346543, A348081.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 27 2021