A099512 Triangle, read by rows, of trinomial coefficients arranged so that there are n+1 terms in row n by setting T(n,k) equal to the coefficient of z^k in (1 + 3*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

1, 1, 3, 1, 6, 1, 1, 9, 11, 6, 1, 12, 30, 45, 1, 1, 15, 58, 144, 30, 9, 1, 18, 95, 330, 195, 144, 1, 1, 21, 141, 630, 685, 873, 58, 12, 1, 24, 196, 1071, 1770, 3258, 685, 330, 1, 1, 27, 260, 1680, 3801, 9198, 3989, 3258, 95, 15, 1, 30, 333, 2484, 7210, 21672, 15533

Offset: 0

Author

Paul D. Hanna, Oct 20 2004

Keywords

• nonn
• tabl

Comments

Row sums form A099513. In general if T(n,k) = coefficient of z^k in (a + b*z + c*z^2)^(n-[k/2]), then the resulting number triangle will have the o.g.f.: ((1-a*x-c*x^2*y^2) + b*x*y)/((1-a*x-c*x^2*y^2)^2 - x*(b*x*y)^2).

Examples

			Rows begin:
[1],
[1,3],
[1,6,1],
[1,9,11,6],
[1,12,30,45,1],
[1,15,58,144,30,9],
[1,18,95,330,195,144,1],
[1,21,141,630,685,873,58,12],
[1,24,196,1071,1770,3258,685,330,1],
[1,27,260,1680,3801,9198,3989,3258,95,15],...
and can be derived from coefficients of (1+3*z+z^2)^n:
[1],
[1,3,1],
[1,6,11,6,1],
[1,9,30,45,30,9,1],
[1,12,58,144,195,144,58,12,1],
[1,15,95,330,685,873,685,330,95,15,1],...
by shifting each column k down by [k/2] rows.
		

Crossrefs

Cf. A099509, A099510, A099513.

Programs

PARI

T(n,k)=if(n
				

Formula

G.f.: (1-x+3*x*y-x^2*y^2)/((1-x)^2-2*x^2*y^2-7*x^3*y^2+x^4*y^4).