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Triangle read by rows: T(n,k)=n*(n+1)*((3*k+2)*n+1)/6, 0<=k<=n.
8

%I #13 Feb 19 2020 05:20:27

%S 0,1,2,5,11,17,14,32,50,68,30,70,110,150,190,55,130,205,280,355,430,

%T 91,217,343,469,595,721,847,140,336,532,728,924,1120,1316,1512,204,

%U 492,780,1068,1356,1644,1932,2220,2508,285,690,1095,1500,1905,2310,2715,3120

%N Triangle read by rows: T(n,k)=n*(n+1)*((3*k+2)*n+1)/6, 0<=k<=n.

%C Row sums give A132122; central terms give A132123

%C T(n,0) = A000330(n);

%C T(n,1) = A033994(n) for n>0;

%C T(n,2) = A132124(n) for n>1;

%C T(n,3) = A132112(n) for n>2;

%C T(n,4) = A050409(n) for n>3.

%F G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k)*x^n*y^k = x*(x*y+1+x)/((1-x)^4*(1-y)^2). - _R. J. Mathar_, Jul 28 2016. Note that this generates a full array, not just the triangular subspace.

%e 0;

%e 1, 2;

%e 5, 11, 17;

%e 14, 32, 50, 68;

%e 30, 70, 110, 150, 190;

%e 55, 130, 205, 280, 355, 430;

%e 91, 217, 343, 469, 595, 721, 847;

%e 140, 336, 532, 728, 924, 1120, 1316, 1512;

%e 204, 492, 780, 1068, 1356, 1644, 1932, 2220, 2508;

%p A132121 := proc(n,k)

%p n*(n+1)*((3*k+2)*n+1)/6 ;

%p end proc:

%p seq(seq(A132121(n,k),k=0..n),n=0..13) ; # _R. J. Mathar_, Feb 19 2020

%K nonn,tabl

%O 0,3

%A _Reinhard Zumkeller_, Aug 12 2007