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A132121 Triangle read by rows: T(n,k)=n*(n+1)*((3*k+2)*n+1)/6, 0<=k<=n.

%I A132121 #18 Mar 04 2025 13:09:08
%S A132121 0,1,2,5,11,17,14,32,50,68,30,70,110,150,190,55,130,205,280,355,430,
%T A132121 91,217,343,469,595,721,847,140,336,532,728,924,1120,1316,1512,204,
%U A132121 492,780,1068,1356,1644,1932,2220,2508,285,690,1095,1500,1905,2310,2715,3120,3525,3930
%N A132121 Triangle read by rows: T(n,k)=n*(n+1)*((3*k+2)*n+1)/6, 0<=k<=n.
%C A132121 Row sums give A132122; central terms give A132123.
%C A132121 T(n,0) = A000330(n);
%C A132121 T(n,1) = A033994(n) for n>0;
%C A132121 T(n,2) = A132124(n) for n>1;
%C A132121 T(n,3) = A132112(n) for n>2;
%C A132121 T(n,4) = A050409(n) for n>3.
%F A132121 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 A132121 0;
%e A132121 1, 2;
%e A132121 5, 11, 17;
%e A132121 14, 32, 50, 68;
%e A132121 30, 70, 110, 150, 190;
%e A132121 55, 130, 205, 280, 355, 430;
%e A132121 91, 217, 343, 469, 595, 721, 847;
%e A132121 140, 336, 532, 728, 924, 1120, 1316, 1512;
%e A132121 204, 492, 780, 1068, 1356, 1644, 1932, 2220, 2508;
%p A132121 A132121 := proc(n,k)
%p A132121     n*(n+1)*((3*k+2)*n+1)/6 ;
%p A132121 end proc:
%p A132121 seq(seq(A132121(n,k),k=0..n),n=0..13) ; # _R. J. Mathar_, Feb 19 2020
%t A132121 Table[n(n+1)((3k+2)n+1)/6,{n,0,9},{k,0,n}]//Flatten (* _James C. McMahon_, Mar 04 2025 *)
%Y A132121 Cf. A132122, A132123.
%Y A132121 Cf. A000330, A033994, A132124, A132112, A050409.
%K A132121 nonn,tabl
%O A132121 0,3
%A A132121 _Reinhard Zumkeller_, Aug 12 2007
%E A132121 a(53)-a(54) from _James C. McMahon_, Mar 04 2025