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
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
Comments