A099527 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 (2 + 3*z + z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2.

1, 2, 3, 4, 12, 1, 8, 36, 13, 6, 16, 96, 66, 63, 1, 32, 240, 248, 360, 33, 9, 64, 576, 800, 1560, 321, 180, 1, 128, 1344, 2352, 5760, 1970, 1683, 62, 12, 256, 3072, 6496, 19152, 9420, 10836, 985, 390, 1, 512, 6912, 17152, 59136, 38472, 55692, 8989, 5418, 100, 15

Offset

0,2

Comments

Row sums form A099528. 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..54.

Formula

G.f.: (1-x*(2-3*y)-x^2*y^2)/(1-4*x+x^2*(4-2*y^2)-5*x^3*y^2+x^4*y^4).

Example

Rows begin:
[1],
[2,3],
[4,12,1],
[8,36,13,6],
[16,96,66,63,1],
[32,240,248,360,33,9],
[64,576,800,1560,321,180,1],
[128,1344,2352,5760,1970,1683,62,12],
[256,3072,6496,19152,9420,10836,985,390,1],
[512,6912,17152,59136,38472,55692,8989,5418,100,15],...
and can be derived from the coefficients of (2+3*z+z^2)^n:
[1],
[2,3,1],
[4,12,13,6,1],
[8,36,66,63,33,9,1],
[16,96,248,360,321,180,62,12,1],
[32,240,800,1560,1970,1683,985,390,100,15,1],...
by shifting each column k down by [k/2] rows.

Prog

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

Crossrefs

Cf. A099509, A099510, A099528.

Sequence in context: A339782 A176621 A370365 * A354371 A345276 A096864

Adjacent sequences: A099524 A099525 A099526 * A099528 A099529 A099530

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 20 2004

Status

approved