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A178821 Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.

%I A178821 #27 Sep 08 2022 08:45:54
%S A178821 1,5,5,15,30,15,35,105,105,35,70,280,420,280,70,126,630,1260,1260,630,
%T A178821 126,210,1260,3150,4200,3150,1260,210,330,2310,6930,11550,11550,6930,
%U A178821 2310,330,495,3960,13860,27720,34650,27720,13860,3960,495,715,6435,25740,60060,90090,90090,60060,25740,6435,715
%N A178821 Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.
%C A178821 The product of the pentatope numbers (A000332, beginning with fifth term) and Pascal's triangle (A007318). Also level 5 of Pascal's prism (A178819) read by rows: (i+4; 4, i-j, j), i >= 0, 0 <= j <= i.
%H A178821 G. C. Greubel, <a href="/A178821/b178821.txt">Rows n=0..100 of triangle, flattened</a>
%H A178821 H. J. Brothers, <a href="https://doi.org/10.1017/S0025557200004447">Pascal's prism</a>, The Mathematical Gazette, 96 (July 2012), 213-220.
%H A178821 H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>
%F A178821 T(n,k) = C(n+4,4) * C(n,k), 0 <= k <= n.
%F A178821 For element a in A178819: a_(5, i, j) = (i+3; 4, i-j, j-1), i >= 1, 1 <= j <= i.
%F A178821 G.f.: 1/(1 - x - x*y)^5. - _Ilya Gutkovskiy_, Mar 20 2020
%e A178821 Triangle begins:
%e A178821    1;
%e A178821    5,   5;
%e A178821   15,  30,  15;
%e A178821   35, 105, 105,  35;
%e A178821   70, 280, 420, 280,  70;
%p A178821 T:=(n,k)->binomial(n+4,4)*binomial(n,k): seq(seq(T(n,k),k=0..n),n=0..9); # _Muniru A Asiru_, Jan 22 2019
%t A178821 Table[Multinomial[4, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
%o A178821 (Magma) /* As triangle */ [[Binomial(n+4,4)*Binomial(n,k): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, Oct 23 2017
%o A178821 (PARI) {T(n,k) = binomial(n+4, 4)*binomial(n, k)}; \\ _G. C. Greubel_, Jan 22 2019
%o A178821 (Sage) [[binomial(n+4, 4)*binomial(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jan 22 2019
%o A178821 (GAP) T:=Flat(List([0..10], n-> List([0..n], k-> Binomial(n+4, 4)* Binomial(n, k) ))); # _G. C. Greubel_, Jan 22 2019
%Y A178821 Cf. A000332, A007318, A178819, A178820, A178822.
%Y A178821 Rows sum to A003472, shallow diagonals sum to A001873.
%K A178821 easy,nonn,tabl
%O A178821 0,2
%A A178821 _Harlan J. Brothers_, Jun 19 2010