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

1, 1, 5, 37, 181, 1301, 9401, 65465, 498037, 3796021, 29221705, 230396585, 1828448425, 14651160265, 118544522045, 965075143037, 7907605360757, 65162569952245, 539515760866889, 4486877961224297, 37463151704756281, 313909383754331801, 2638892573249746445, 22249830926517611917

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

• nonn

Comments

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y)^2*z*w).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1053

Crossrefs

Cf. A000984, A002426, A005259, A005260, A051286, A181546, A275027, A277247, A382842.

Cf. A089627.

Programs

Maple

a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^2, k=0..n/2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 07 2025

Mathematica

Table[Sum[(Binomial[n, k] Binomial[n - k, k])^2, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1, 1}, 16], {n, 0, 23}]
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}]

Formula

a(n) ~ 3^(2*n+2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 07 2025

a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^2. - Alois P. Heinz, Apr 07 2025