A321500 - OEIS

%I #23 Sep 08 2022 08:46:23

%S 0,1,4,8,15,32,27,40,65,108,64,85,120,175,256,125,156,203,272,369,500,

%T 216,259,320,405,520,671,864,343,400,477,580,715,888,1105,1372,512,

%U 585,680,803,960,1157,1400,1695,2048

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

%H M. F. Hasler, <a href="/A321500/b321500.txt">Rows n = 0..141 of triangle, flattened</a>

%F 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

%e The table starts:

%e n | T(n,k), k = 0..n:

%e 0 |   0;

%e 1 |   1,   4;

%e 2 |   8,  15,  32;

%e 3 |  27,  40,  65, 108;

%e 4 |  64,  85, 120, 175, 256;

%e 5 | 125, 156, 203, 272, 369,  500;

%e 6 | 216, 259, 320, 405, 520,  671,  864;

%e 7 | 343, 400, 477, 580, 715,  888, 1105, 1372;

%e 8 | 512, 585, 680, 803, 960, 1157, 1400, 1695, 2048;

%e etc.

%t 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 *)

%o (PARI) A321500(n, k)=(n+k)*(n^2+k^2)

%o A321500_row(n)=vector(n+1, k, (n+k--)*(n^2+k^2))

%o A321500_list(N=11)=concat(apply(A321500_row, [0..N]))

%o (Magma) [[(n+k)*(n^2+k^2): k in [0..n]]: n in [0..12]]; // _G. C. Greubel_, Nov 23 2018

%o (Sage) [[(n+k)*(n^2+k^2) for k in range(n+1)] for n in range(12)] # _G. C. Greubel_, Nov 23 2018

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

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

%K nonn,tabl,easy

%O 0,3

%A _M. F. Hasler_, Nov 22 2018