A321500 Triangular table T(n,k) = (n+k)*(n^2+k^2), n >= k >= 0; read by rows n = 0, 1, 2, ...

0, 1, 4, 8, 15, 32, 27, 40, 65, 108, 64, 85, 120, 175, 256, 125, 156, 203, 272, 369, 500, 216, 259, 320, 405, 520, 671, 864, 343, 400, 477, 580, 715, 888, 1105, 1372, 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048

Offset: 0

Author

M. F. Hasler, Nov 22 2018

Keywords

• nonn
• tabl
• easy

Examples

			The table starts:
n | T(n,k), k = 0..n:
0 |   0;
1 |   1,   4;
2 |   8,  15,  32;
3 |  27,  40,  65, 108;
4 |  64,  85, 120, 175, 256;
5 | 125, 156, 203, 272, 369,  500;
6 | 216, 259, 320, 405, 520,  671,  864;
7 | 343, 400, 477, 580, 715,  888, 1105, 1372;
8 | 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048;
etc.
		

Links

• M. F. Hasler, Rows n = 0..141 of triangle, flattened

Crossrefs

Cf. A000578 (column 0: the cubes), A033430 (diagonal: 4*n^3), A053698 (column 1).

Cf. A198063 (read as A(n,k)=(n+k)*(n^2+k^2)).

Programs

Magma

[[(n+k)*(n^2+k^2): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Nov 23 2018

Mathematica

t[n_, k_] := (n + k) (n^2 + k^2); Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Nov 22 2018 *)

PARI

A321500(n, k)=(n+k)*(n^2+k^2)
A321500_row(n)=vector(n+1, k, (n+k--)*(n^2+k^2))
A321500_list(N=11)=concat(apply(A321500_row, [0..N]))

Sage

[[(n+k)*(n^2+k^2) for k in range(n+1)] for n in range(12)] # G. C. Greubel, Nov 23 2018

Formula

Sum_{k=0..n} T(n,k) = 5*n^2*(n+1)*(5*n+1)/12 = 5*A117066(n). - G. C. Greubel, Nov 23 2018