A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

1, 4, 4, 10, 20, 10, 20, 60, 60, 20, 35, 140, 210, 140, 35, 56, 280, 560, 560, 280, 56, 84, 504, 1260, 1680, 1260, 504, 84, 120, 840, 2520, 4200, 4200, 2520, 840, 120, 165, 1320, 4620, 9240, 11550, 9240, 4620, 1320, 165, 220, 1980, 7920, 18480, 27720, 27720, 18480, 7920, 1980, 220

Offset: 0

Author

Harlan J. Brothers, Jun 17 2010

Keywords

• easy
• nonn
• tabl

Comments

The product of the tetrahedral numbers (A000292, beginning with second term) and Pascal's triangle (A007318). Also level 4 of Pascal's prism (A178819): (i+3; 3, i-j, j), i >= 0, 0 <= j <= i.

Examples

			Triangle begins:
   1;
   4,   4;
  10,  20,  10;
  20,  60,  60,  20;
  35, 140, 210, 140,  35;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened
• H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
• H. J. Brothers, Pascal's Prism: Supplementary Material

Crossrefs

Cf. A000292, A007318, A178819.

Rows sums give A001789.

Programs

GAP

T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+3, 3)* Binomial(n, k) ))); # G. C. Greubel, Jan 22 2019

Magma

/* As triangle */ [[Binomial(n+3,3)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Oct 23 2017

Maple

T:=(n,k)->binomial(n+3,3)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Jan 22 2019

Mathematica

Table[Multinomial[3, i-j, j], {i, 0, 9}, {j, 0, i}]//Column

PARI

{T(n,k) = binomial(n+3, 3)*binomial(n, k)}; \\ G. C. Greubel, Jan 22 2019

Sage

[[binomial(n+3, 3)*binomial(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Jan 22 2019

Formula

T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.

For element a in A178819: a_(4, i, j) = (i+2; 3, i-j, j-1), i >= 1, 1 <= j <= i.

G.f.: 1/(1 - x - x*y)^4. - Ilya Gutkovskiy, Mar 20 2020