A277188 The binomial sum a(n) = Sum_{k=0..n}(binomial(n,k)*binomial(n+1,k+1)*binomial(n+2,k+2)).

1, 7, 43, 281, 1896, 13112, 92359, 659941, 4769758, 34797170, 255838760, 1893389720, 14091400480, 105385445856, 791504226943, 5966958725021, 45133376297922, 342400478465678, 2604549070175770, 19860078537996958, 151769147958738016

Offset: 0

Author

Emanuele Munarini, Oct 04 2016

Keywords

• nonn

Programs

Mathematica

Table[Sum[Binomial[n,k]Binomial[n+1,k+1]Binomial[n+2,k+2],{k,0,n}],{n,0,100}]

Maxima

makelist((n+1)^2*(n+2)/2*hypergeometric([-n,-n,-n],[2,3],-1),n,0,12);

PARI

a(n) = sum(k=0, n, (binomial(n,k)*binomial(n+1,k+1)*binomial(n+2,k+2))); \\ Michel Marcus, Oct 04 2016

Formula

a(n) = (1/2)*(n+1)^2*(n+2)*hypergeometric({-n,-n,-n},{2,3},-1).

From Vaclav Kotesovec, Oct 04 2016: (Start)

Recurrence: (n+2)^2*(6*n^3 + 3*n^2 - 3*n - 2)*a(n) = (42*n^5 + 147*n^4 + 147*n^3 - 8*n^2 - 60*n - 16)*a(n-1) + 8*(n-1)*n*(6*n^3 + 21*n^2 + 21*n + 4)*a(n-2).

a(n) ~ 2^(3*n+4)/(sqrt(3)*Pi*n).

(End)