A116088 Riordan array (1, x*(1+x)^2).

1, 0, 1, 0, 2, 1, 0, 1, 4, 1, 0, 0, 6, 6, 1, 0, 0, 4, 15, 8, 1, 0, 0, 1, 20, 28, 10, 1, 0, 0, 0, 15, 56, 45, 12, 1, 0, 0, 0, 6, 70, 120, 66, 14, 1, 0, 0, 0, 1, 56, 210, 220, 91, 16, 1, 0, 0, 0, 0, 28, 252, 495, 364, 120, 18, 1

Offset

0,5

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (Rows 0 <= n <= 150).

Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Formula

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

Number triangle T(n,k) = C(2*k, n-k) = C(n,k)*C(3*k,n)/C(3*k,k).

Example

Triangle begins as:
  1;
  0, 1;
  0, 2, 1;
  0, 1, 4,  1;
  0, 0, 6,  6,  1;
  0, 0, 4, 15,  8,  1;
  0, 0, 1, 20, 28, 10,  1;
  0, 0, 0, 15, 56, 45, 12, 1;

Mathematica

Flatten[Table[Binomial[2k, n-k], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Oct 22 2012 *)

Prog

(PARI) {T(n, k) = binomial(2*k, n-k)}; \\ G. C. Greubel, May 09 2019
(Magma) [[Binomial(2*k, n-k): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 09 2019
(Sage) [[binomial(2*k, n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 09 2019
(GAP) Flat(List([0..10], n->List([0..n], k-> Binomial(2*k, n-k) ))); # G. C. Greubel, May 09 2019

Crossrefs

Row sums are A002478. Diagonal sums are A094686. Inverse is (-1)^(n-k) * A109971(n,k). Unsigned version of A109970.

Sequence in context: A287698 A366834 A109970 * A136501 A180983 A127709

Adjacent sequences: A116085 A116086 A116087 * A116089 A116090 A116091

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 04 2006

Status

approved