A262407 a(n) = Sum_{k=0..n-1} C(n,k+1)*C(n,k)*C(n-1,k).

0, 1, 4, 24, 152, 1010, 6912, 48328, 343408, 2471274, 17966360, 131717960, 972488640, 7223061040, 53925450880, 404400203280, 3044645475296, 23002424245754, 174324246314184, 1324800580881952, 10093304926771600, 77073430602848316, 589761299099196224

Offset: 0

Author

M. F. Hasler, Sep 21 2015

Keywords

• nonn

Crossrefs

Cf. A000489, A000535, A000279.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n,
       ((21*n^3-49*n^2+30*n-8)*a(n-1)+
        (8*(n-1))*(n-2)*(3*n-1)*a(n-2))/
        ((3*n-4)*(n+1)*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 22 2015

Mathematica

f[n_]:=HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]; a[n_]:=n^2 (f[n] + 4 f[n - 1])/(3 n^2 + 3 n); Array[a, 25] (* Vincenzo Librandi, Sep 22 2015 *)
Table[Sum[Binomial[n,k+1]Binomial[n,k]Binomial[n-1,k],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Apr 09 2021 *)

PARI

a(n)=sum(k=0, n-1, binomial(n, k+1)*binomial(n, k)*binomial(n-1, k))

Formula

a(n) = A000279(n)/(3*n) = (A000172(n)+4*A000172(n-1))*n/(3*(n+1)).

a(n) ~ 8^n/(sqrt(3)*Pi*n) as n -> oo.