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A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.

%I A178822 #25 Sep 08 2022 08:45:54
%S A178822 1,6,6,21,42,21,56,168,168,56,126,504,756,504,126,252,1260,2520,2520,
%T A178822 1260,252,462,2772,6930,9240,6930,2772,462,792,5544,16632,27720,27720,
%U A178822 16632,5544,792,1287,10296,36036,72072,90090,72072,36036,10296,1287
%N A178822 Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
%C A178822 The product of A000389 and Pascal's triangle (A007318). Level 6 of Pascal's prism (A178819) read by rows: (i+5; 5, i-j, j), i >= 0, 0 <= j <= i.
%H A178822 G. C. Greubel, <a href="/A178822/b178822.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A178822 H. J. Brothers, <a href="https://doi.org/10.1017/S0025557200004447">Pascal's prism</a>, The Mathematical Gazette, 96 (July 2012), 213-220.
%H A178822 H. J. Brothers, <a href="http://www.brotherstechnology.com/math/pascals-prism.html">Pascal's Prism: Supplementary Material</a>
%F A178822 T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.
%F A178822 For element a_(h, i, j) in A178819: a_(6, i, j) = (i+4; 5, i-j, j-1), i >= 1, 1 <= j <= i.
%F A178822 G.f.: 1/(1 - x - x*y)^6. - _Ilya Gutkovskiy_, Mar 20 2020
%e A178822 Triangle begins:
%e A178822     1;
%e A178822     6,   6;
%e A178822    21,  42,  21;
%e A178822    56, 168, 168,  56;
%e A178822   126, 504, 756, 504, 126;
%t A178822 Table[Multinomial[5, i-j, j], {i, 0, 9}, {j, 0, i}]//Column
%t A178822 Table[Binomial[n + 5, 5]*Binomial[n, k], {n,0,10}, {k,0,n}] // Flatten (* _G. C. Greubel_, Nov 25 2017 *)
%o A178822 (Magma) /* As triangle */ [[Binomial(n+5,5)*Binomial(n,k): k in [0..n]]: n in [0..10]]; // _Vincenzo Librandi_, Oct 23 2017
%o A178822 (PARI) for(n=0,10, for(k=0,n, print1(binomial(n+5,5)*binomial(n,k), ", "))) \\ _G. C. Greubel_, Nov 25 2017
%Y A178822 Cf. A000389, A007318, A178819, A178820, A178821.
%Y A178822 Rows sum to A054849, shallow diagonals sum to A001874.
%K A178822 easy,nonn,tabl
%O A178822 0,2
%A A178822 _Harlan J. Brothers_, Jun 19 2010