A381482 a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)^2 * binomial(n-k,k) * 2^k.

1, 1, 9, 37, 241, 1401, 8961, 57429, 377217, 2509201, 16876729, 114600069, 783903121, 5397915433, 37372017489, 259998843477, 1816376953857, 12736545070113, 89602978644969, 632223913939557, 4472680961409201, 31717890254271321, 225416254500886689, 1605197563027768917

Offset: 0

Author

Ilya Gutkovskiy, Apr 22 2025

Keywords

• nonn

Comments

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

Crossrefs

Cf. A001045, A001850, A084601, A206178, A275027.

Programs

Mathematica

Table[Sum[Binomial[n, k]^2 Binomial[n - k, k] 2^k, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, -n, -n/2}, {1, 1}, -8], {n, 0, 23}]

Formula

a(n) = hypergeom( [1/2 - n/2, -n, -n/2], [1, 1], -8).

a(n) ~ sqrt(7/12 + sqrt(89/38)*cosh(arccosh((8567*sqrt(19/178))/1424)/3)/3) * ((1/3 + 8*sqrt(7)*(cosh(arccosh(1261/(448*sqrt(7)))/3)/3))^n / Pi) / n. - Vaclav Kotesovec, Apr 23 2025