A116089 Riordan array (1, x*(1+x)^3).

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 1, 15, 9, 1, 0, 0, 20, 36, 12, 1, 0, 0, 15, 84, 66, 15, 1, 0, 0, 6, 126, 220, 105, 18, 1, 0, 0, 1, 126, 495, 455, 153, 21, 1, 0, 0, 0, 84, 792, 1365, 816, 210, 24, 1, 0, 0, 0, 36, 924, 3003, 3060, 1330, 276, 27, 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)^3).

Number triangle T(n,k) = C(3*k,n-k) = C(n,k)*C(4*k,n)/C(4*k,k).

Example

Triangle begins as:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 1, 15,   9,   1;
  0, 0, 20,  36,  12,    1;
  0, 0, 15,  84,  66,   15,    1;
  0, 0,  6, 126, 220,  105,   18,    1;
  0, 0,  1, 126, 495,  455,  153,   21,   1;
  0, 0,  0,  84, 792, 1365,  816,  210,  24,  1;
  0, 0,  0,  36, 924, 3003, 3060, 1330, 276, 27, 1;

Mathematica

Flatten[Table[Binomial[3k, n-k], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Feb 05 2012 *)

Prog

(PARI) {T(n, k) = binomial(3*k, n-k)}; \\ G. C. Greubel, May 09 2019
(Magma) [[Binomial(3*k, n-k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, May 09 2019
(Sage) [[binomial(3*k, n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, May 09 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(3*k, n-k) ))); # G. C. Greubel, May 09 2019

Crossrefs

Row sums are A099234. Diagonal sums are A116090.

Sequence in context: A292727 A049403 A104556 * A122016 A067882 A215771

Adjacent sequences: A116086 A116087 A116088 * A116090 A116091 A116092

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 04 2006

Status

approved