A108478 Diagonal sums of number triangle A108477.

1, 1, 2, 14, 43, 127, 468, 1596, 5253, 17917, 60918, 205194, 694287, 2351611, 7951336, 26894840, 91004105, 307854073, 1041410602, 3523170438, 11918842803, 40320750711, 136404504124, 461454010164, 1561085306061, 5281113937653

Offset: 0

Author

Paul Barry, Jun 04 2005

Keywords

• easy
• nonn

Programs

Maple

A108478:=n->add(add(binomial(2*(n-2*k),j)*binomial(2*k,j)*2^j, j=0..n-k), k=0..floor(n/2)): seq(A108478(n), n=0..30); # Wesley Ivan Hurt, Sep 26 2014

Mathematica

Table[Sum[Sum[Binomial[2 (n - 2 k), j]*Binomial[2 k, j]*2^j, {j, 0, n - k}], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Wesley Ivan Hurt, Sep 26 2014 *)

PARI

a(n) = sum(k=0, n\2, sum(j=0, n-k, binomial(2*(n-2*k), j)*binomial(2*k, j)*2^j)); \\ Michel Marcus, Sep 26 2014

Formula

a(n) = sum_{k=0..floor(n/2)} ( sum_{j=0..n-k} C(2(n-2k), j)*C(2k, j)*2^j ).

Empirical g.f.: -(3*x^3+x^2+x-1) / ((x^3-3*x^2-x-1)*(x^3+x^2+3*x-1)). - Colin Barker, Sep 26 2014