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

0,3

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).

Links

Table of n, a(n) for n=0..61.

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).

Example

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.

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, polcoeff((1+3*z+z^2+z*O(z^k))^(n-k\2), k, z))

Crossrefs

Cf. A099509, A099510, A099513.

Sequence in context: A116609 A124846 A177375 * A127096 A130541 A128489

Adjacent sequences: A099509 A099510 A099511 * A099513 A099514 A099515

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 20 2004

Status

approved