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A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

%I A382841 #15 Apr 15 2025 15:10:08
%S A382841 1,1,5,37,181,1301,9401,65465,498037,3796021,29221705,230396585,
%T A382841 1828448425,14651160265,118544522045,965075143037,7907605360757,
%U A382841 65162569952245,539515760866889,4486877961224297,37463151704756281,313909383754331801,2638892573249746445,22249830926517611917
%N A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.
%C A382841 Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y)^2*z*w).
%H A382841 Alois P. Heinz, <a href="/A382841/b382841.txt">Table of n, a(n) for n = 0..1053</a>
%F A382841 a(n) ~ 3^(2*n+2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Apr 07 2025
%F A382841 a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^2. - _Alois P. Heinz_, Apr 07 2025
%p A382841 a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^2, k=0..n/2):
%p A382841 seq(a(n), n=0..23);  # _Alois P. Heinz_, Apr 07 2025
%t A382841 Table[Sum[(Binomial[n, k] Binomial[n - k, k])^2, {k, 0, Floor[n/2]}], {n, 0, 23}]
%t A382841 Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1, 1}, 16], {n, 0, 23}]
%t A382841 Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - w) - (x y)^2 z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 23}]
%Y A382841 Cf. A000984, A002426, A005259, A005260, A051286, A181546, A275027, A277247, A382842.
%Y A382841 Cf. A089627.
%K A382841 nonn
%O A382841 0,3
%A A382841 _Ilya Gutkovskiy_, Apr 06 2025