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

0, 1, 2, 5, 11, 17, 14, 32, 50, 68, 30, 70, 110, 150, 190, 55, 130, 205, 280, 355, 430, 91, 217, 343, 469, 595, 721, 847, 140, 336, 532, 728, 924, 1120, 1316, 1512, 204, 492, 780, 1068, 1356, 1644, 1932, 2220, 2508, 285, 690, 1095, 1500, 1905, 2310, 2715, 3120, 3525, 3930

Offset: 0

Author

Reinhard Zumkeller, Aug 12 2007

Keywords

• nonn
• tabl

Comments

Row sums give A132122; central terms give A132123.

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

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

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

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

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

Examples

			0;
1, 2;
5, 11, 17;
14, 32, 50, 68;
30, 70, 110, 150, 190;
55, 130, 205, 280, 355, 430;
91, 217, 343, 469, 595, 721, 847;
140, 336, 532, 728, 924, 1120, 1316, 1512;
204, 492, 780, 1068, 1356, 1644, 1932, 2220, 2508;
		

Crossrefs

Cf. A132122, A132123.

Cf. A000330, A033994, A132124, A132112, A050409.

Programs

Maple

A132121 := proc(n,k)
    n*(n+1)*((3*k+2)*n+1)/6 ;
end proc:
seq(seq(A132121(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Feb 19 2020

Mathematica

Table[n(n+1)((3k+2)n+1)/6,{n,0,9},{k,0,n}]//Flatten (* James C. McMahon, Mar 04 2025 *)

Formula

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.

Extensions

a(53)-a(54) from James C. McMahon, Mar 04 2025