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

%I A178820 #26 Sep 08 2022 08:45:54
%S A178820 1,4,4,10,20,10,20,60,60,20,35,140,210,140,35,56,280,560,560,280,56,
%T A178820 84,504,1260,1680,1260,504,84,120,840,2520,4200,4200,2520,840,120,165,
%U A178820 1320,4620,9240,11550,9240,4620,1320,165,220,1980,7920,18480,27720,27720,18480,7920,1980,220
%N A178820 Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
%C A178820 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 A178820 G. C. Greubel, <a href="/A178820/b178820.txt">Rows n=0..100 of triangle, flattened</a>
%H A178820 H. J. Brothers, <a href="https://doi.org/10.1017/S0025557200004447">Pascal's prism</a>, The Mathematical Gazette, 96 (July 2012), 213-220.
%H A178820 H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>
%F A178820 T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.
%F A178820 For element a in A178819: a_(4, i, j) = (i+2; 3, i-j, j-1), i >= 1, 1 <= j <= i.
%F A178820 G.f.: 1/(1 - x - x*y)^4. - _Ilya Gutkovskiy_, Mar 20 2020
%e A178820 Triangle begins:
%e A178820    1;
%e A178820    4,   4;
%e A178820   10,  20,  10;
%e A178820   20,  60,  60,  20;
%e A178820   35, 140, 210, 140,  35;
%p A178820 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 A178820 Table[Multinomial[3, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
%o A178820 (Magma) /* As triangle */ [[Binomial(n+3,3)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Oct 23 2017
%o A178820 (PARI) {T(n,k) = binomial(n+3, 3)*binomial(n, k)}; \\ _G. C. Greubel_, Jan 22 2019
%o A178820 (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 A178820 (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 A178820 Cf. A000292, A007318, A178819.
%Y A178820 Rows sums give A001789.
%K A178820 easy,nonn,tabl
%O A178820 0,2
%A A178820 _Harlan J. Brothers_, Jun 17 2010