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A298851 a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x).

%I A298851 #53 May 15 2025 10:56:18
%S A298851 1,1,21,1408,196053,46587905,16875270660,8657594647800,
%T A298851 5974284925007685,5336898188553325075,5992171630749371157181,
%U A298851 8260051854943114812198756,13714895317396748230146099660,26998129079190909699998105620908,62173633286588800021263427046090792
%N A298851 a(n) = [x^n] Product_{k=1..n} 1/(1-k^2*x).
%H A298851 Vaclav Kotesovec, <a href="/A298851/b298851.txt">Table of n, a(n) for n = 0..200</a>
%F A298851 From _Vaclav Kotesovec_, Feb 02 2018, updated May 12 2025: (Start)
%F A298851 a(n) ~ c * d^n * n^(2*n - 1/2), where d = 1.774513671664430848697327843228386312953174421074432567764556466987... and c = 0.617929515483613293691991371141292259390065108300160936187723552669...
%F A298851 In closed form, a(n) ~ n^(2*n - 1/2) * r^(4*n + 1) / (sqrt(Pi*(2 - r^2)) * (r^2 - 1)^n * exp(2*n)), where r = 1.04438203376083348498401390634474776086902815721... is the root of the equation (1-r)/(1+r) = -exp(-4/r). (End)
%F A298851 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 A298851 a(n) = A036969(2n,n) = A269945(2n,n). - _Alois P. Heinz_, Feb 19 2022
%F A298851 From _Seiichi Manyama_, May 12 2025: (Start)
%F A298851 a(n) = Sum_{k=0..2*n} (-n)^k * binomial(4*n,k) * Stirling2(4*n-k,2*n).
%F A298851 a(n) = Sum_{k=0..2*n} (-1)^k * Stirling2(k+n,n) * Stirling2(3*n-k,n). (End)
%p A298851 b:= proc(k, n) option remember; `if`(k=0, 1,
%p A298851       add(b(k-1, j)*j^2, j=1..n))
%p A298851     end:
%p A298851 a:= n-> b(n$2):
%p A298851 seq(a(n), n=0..14);  # _Alois P. Heinz_, Feb 19 2022
%t A298851 Table[SeriesCoefficient[Product[1/(1 - k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 02 2018 *)
%t A298851 Join[{1}, Table[2*Sum[(-1)^(n-k) * Binomial[2*n, n-k] * k^(4*n), {k, 0, n}]/(2*n)!, {n, 1, 20}]] (* _Vaclav Kotesovec_, May 15 2025 *)
%o A298851 (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 A298851 makelist(a(n), n, 0, 20); /* _Tani Akinari_, Mar 09 2021 */
%Y A298851 Cf. A001044, A036969, A269945.
%Y A298851 Cf. A007820, A299035, A299036.
%Y A298851 Cf. A383853.
%K A298851 nonn
%O A298851 0,3
%A A298851 _Seiichi Manyama_, Feb 01 2018