A143180 Triangle read by rows: T(n, k) = (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1).

-1, 3, 3, 19, 4, 19, 53, 17, 17, 53, 111, 48, 27, 48, 111, 199, 103, 55, 55, 103, 199, 323, 188, 107, 80, 107, 188, 323, 489, 309, 189, 129, 129, 189, 309, 489, 703, 472, 307, 208, 175, 208, 307, 472, 703, 971, 683, 467, 323, 251, 251, 323, 467, 683, 971, 1299, 948, 675, 480, 363, 324, 363, 480, 675, 948, 1299

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Oct 17 2008

Keywords

• sign
• tabl

Examples

			Triangle begins as:
    -1;
     3,   3;
    19,   4,  19;
    53,  17,  17,  53;
   111,  48,  27,  48, 111;
   199, 103,  55,  55, 103, 199;
   323, 188, 107,  80, 107, 188, 323;
   489, 309, 189, 129, 129, 189, 309, 489;
   703, 472, 307, 208, 175, 208, 307, 472, 703;
   971, 683, 467, 323, 251, 251, 323, 467, 683, 971;
  1299, 948, 675, 480, 363, 324, 363, 480, 675, 948, 1299;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A295709.

Programs

Magma

A143180:= func< n,k | (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1) >;
[A143180(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 19 2024

Mathematica

T[n_,k_]:= (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1);
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A143180(n,k): return (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1)
flatten([[A143180(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 19 2024

Formula

T(n, k) = (n-k+1)^3 + (k+1)^3 - 3*(n-k+1)*(k+1).

T(n, n-k) = T(n, k).

G.f.: ((-1 + 7*x + x^2 - x^3) + (7 - 36*x + 15*x^2 - 4*x^3)*(x*y) + (1 + 15*x + 3*x^2 - x^3)*(x*y)^2 - (1 + 4*x + x^2)*(x*y)^3)/((1-x)^4*(1 - x*y)^4). - G. C. Greubel, Apr 22 2024

Extensions

Edited by G. C. Greubel, Apr 19 2024