login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099514 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 + z + 2*z^2)^(n-[k/2]), for n>=k>=0, where [k/2] is the integer floor of k/2. 1
1, 1, 1, 1, 2, 2, 1, 3, 5, 4, 1, 4, 9, 13, 4, 1, 5, 14, 28, 18, 12, 1, 6, 20, 50, 49, 56, 8, 1, 7, 27, 80, 105, 161, 56, 32, 1, 8, 35, 119, 195, 366, 210, 200, 16, 1, 9, 44, 168, 329, 72, 1, 581, 732, 160, 80, 1, 10, 54, 228, 518, 1288, 1337, 2045, 780, 640, 32, 1, 11, 65, 300 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums form A099515. 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..70.

FORMULA

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

EXAMPLE

Rows begin:

[1],

[1,1],

[1,2,2],

[1,3,5,4],

[1,4,9,13,4],

[1,5,14,28,18,12],

[1,6,20,50,49,56,8],

[1,7,27,80,105,161,56,32],

[1,8,35,119,195,366,210,200,16],

[1,9,44,168,329,721,581,732,160,80],...

and can be derived from coefficients of (1+z+2*z^2)^n:

[1],

[1,1,2],

[1,2,5,4,4],

[1,3,9,13,18,12,8],

[1,4,14,28,49,56,56,32,16],

[1,5,20,50,105,161,210,200,160,80,32],...

by shifting each column k down by [k/2] rows.

PROG

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

CROSSREFS

Cf. A099509, A099510, A099512, A099515.

Sequence in context: A257005 A160232 A026300 * A228352 A205575 A257006

Adjacent sequences:  A099511 A099512 A099513 * A099515 A099516 A099517

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 20 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 23 05:44 EST 2018. Contains 299473 sequences. (Running on oeis4.)