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A269956 Triangle read by rows, T(n,k) = binomial(3*n,n+k) for n>=0 and 0<=k<=n.

%I A269956 #17 Sep 08 2022 08:46:16
%S A269956 1,3,3,15,20,15,84,126,126,84,495,792,924,792,495,3003,5005,6435,6435,
%T A269956 5005,3003,18564,31824,43758,48620,43758,31824,18564,116280,203490,
%U A269956 293930,352716,352716,293930,203490,116280,735471,1307504,1961256,2496144,2704156,2496144,1961256,1307504,735471
%N A269956 Triangle read by rows, T(n,k) = binomial(3*n,n+k) for n>=0 and 0<=k<=n.
%H A269956 Felix Fröhlich, <a href="/A269956/b269956.txt">Table of n, a(n) for n = 0..10000</a>
%F A269956 T(n,0) = T(n,n) = A005809(n).
%e A269956 Triangle starts:
%e A269956 1,
%e A269956 3,    3,
%e A269956 15,   20,   15,
%e A269956 84,   126,  126,  84,
%e A269956 495,  792,  924,  792,  495,
%e A269956 3003, 5005, 6435, 6435, 5005, 3003
%p A269956 for n from 0 to 6 do seq(binomial(3*n,n+k), k=0..n) od;
%t A269956 Table[Binomial[3 n, n + k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Vincenzo Librandi_, Mar 28 2016 *)
%o A269956 (Magma) /* As triangle */ [[Binomial(3*n, n+k): k in [0..n]]: n in [0.. 7]]; // _Vincenzo Librandi_, Mar 28 2016
%o A269956 (PARI) t(n, k) = binomial(3*n, n+k)
%o A269956 trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
%o A269956 /* The following function call prints the upper six rows of the triangle */
%o A269956 trianglerows(6) \\ _Felix Fröhlich_, Mar 28 2016
%Y A269956 Cf. binomial(m*n,n+k): A000007 (m=0), A010054 (m=1), A094527 (m=2).
%Y A269956 Cf. A005809.
%K A269956 nonn,tabl
%O A269956 0,2
%A A269956 _Peter Luschny_, Mar 28 2016