A109955 Number triangle binomial(n+2k,3k).

1, 1, 1, 1, 4, 1, 1, 10, 7, 1, 1, 20, 28, 10, 1, 1, 35, 84, 55, 13, 1, 1, 56, 210, 220, 91, 16, 1, 1, 84, 462, 715, 455, 136, 19, 1, 1, 120, 924, 2002, 1820, 816, 190, 22, 1, 1, 165, 1716, 5005, 6188, 3876, 1330, 253, 25, 1, 1, 220, 3003, 11440, 18564, 15504, 7315, 2024

Offset: 0

Author

Paul Barry, Jul 06 2005

Keywords

• easy
• nonn
• tabl

Comments

Riordan array (1/(1-x),x/(1-x)^3).

Inverse array is A109956. Row sums are A052544.

Diagonal sums are A034943(n+1).

Examples

			Rows begin
1;
1,1;
1,4,1;
1,10,7,1;
1,20,28,10,1;
1,35,84,55,13,1;
		

Programs

PARI

tabl(nn) = {my(m = matrix(nn, nn, n, k, if (nMichel Marcus, Nov 20 2015

Formula

Number triangle T(n, k) = binomial(n+2k, 3k).

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k) + T(n-1,k-1). - Philippe Deléham, Feb 18 2012

G.f.: (1-x)^2/((1-x)^3-y*x). - Philippe Deléham, Feb 18 2012

Sum_{k, 0<=k<=n} T(n,k)*x^k = A185963(n), A000012(n), A052544(n), A049086(n) for x = -1, 0, 1, 2 respectively. - Philippe Deléham, Feb 18 2012