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%I #38 Feb 19 2022 10:16:01
%S 1,1,21,1408,196053,46587905,16875270660,8657594647800,
%T 5974284925007685,5336898188553325075,5992171630749371157181,
%U 8260051854943114812198756,13714895317396748230146099660,26998129079190909699998105620908,62173633286588800021263427046090792
%N a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x).
%H Vaclav Kotesovec, <a href="/A298851/b298851.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.774513671664430848697327843228386312953174421074432567764556466987... and c = 0.617929515483613293691991371141292259390065108300160936187723552669... - _Vaclav Kotesovec_, Feb 02 2018
%F a(n) = 2*(Sum_{k=0..n} (n-k)^(4*n)/((2*n-k)!*k!*(-1)^k)) for n>0 - _Tani Akinari_, Mar 09 2021
%F a(n) = A036969(2n,n) = A269945(2n,n). - _Alois P. Heinz_, Feb 19 2022
%p b:= proc(k, n) option remember; `if`(k=0, 1,
%p add(b(k-1, j)*j^2, j=1..n))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..14); # _Alois P. Heinz_, Feb 19 2022
%t Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 02 2018 *)
%o (Maxima) a(n):=if n<1 then 1 else 2*sum((n-k)^(4*n)/((2*n-k)!*k!*(-1)^k),k,0,n);
%o makelist(a(n), n, 0, 20); /* _Tani Akinari_, Mar 09 2021 */
%Y Cf. A001044, A036969, A269945.
%Y Cf. A007820, A299035, A299036.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Feb 01 2018
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