A346183 a(n) = Sum_{k=0..n} binomial(n,k) * binomial((k+1)^2, n).

1, 5, 48, 824, 20690, 687582, 28488488, 1415047216, 81971972604, 5426378062364, 404122795201488, 33442612446777888, 3044479614669988040, 302377373253190949560, 32537275691504428919040, 3770760967834168275347072, 468240057706224000130749072, 62024410203403175896065018192

Offset: 0

Author

Vaclav Kotesovec, Jul 09 2021

Keywords

• nonn

Crossrefs

Cf. A003236.

Programs

Mathematica

Table[Sum[Binomial[n, k]*Binomial[(k+1)^2, n], {k, 0, n}], {n, 0, 20}]

PARI

a(n) = sum(k=0, n, binomial(n,k) * binomial((k+1)^2, n)); \\ Michel Marcus, Jul 09 2021

Formula

a(n) ~ 2^(2*n) * n^(n - 1/2) / (sqrt(Pi*(1+c)) * c^(n + 1/2) * (2+c)^n * exp(n - 1/2 + c^2/8)), where c = LambertW(2*exp(-2)) = 0.21771510575709011079475830443...