%I #26 Sep 08 2022 08:45:54

%S 1,4,4,10,20,10,20,60,60,20,35,140,210,140,35,56,280,560,560,280,56,

%T 84,504,1260,1680,1260,504,84,120,840,2520,4200,4200,2520,840,120,165,

%U 1320,4620,9240,11550,9240,4620,1320,165,220,1980,7920,18480,27720,27720,18480,7920,1980,220

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

%C 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.

%H G. C. Greubel, <a href="/A178820/b178820.txt">Rows n=0..100 of triangle, flattened</a>

%H H. J. Brothers, <a href="https://doi.org/10.1017/S0025557200004447">Pascal's prism</a>, The Mathematical Gazette, 96 (July 2012), 213-220.

%H H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>

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

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

%F G.f.: 1/(1 - x - x*y)^4. - _Ilya Gutkovskiy_, Mar 20 2020

%e Triangle begins:

%e 1;

%e 4, 4;

%e 10, 20, 10;

%e 20, 60, 60, 20;

%e 35, 140, 210, 140, 35;

%p 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

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

%o (Magma) /* As triangle */ [[Binomial(n+3,3)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Oct 23 2017

%o (PARI) {T(n,k) = binomial(n+3, 3)*binomial(n, k)}; \\ _G. C. Greubel_, Jan 22 2019

%o (Sage) [[binomial(n+3, 3)*binomial(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 22 2019

%o (GAP) T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+3, 3)* Binomial(n, k) ))); # _G. C. Greubel_, Jan 22 2019

%Y Cf. A000292, A007318, A178819.

%Y Rows sums give A001789.

%K easy,nonn,tabl

%O 0,2

%A _Harlan J. Brothers_, Jun 17 2010