A129367 Triangle T(n, k) = A002415(n-k+3)*A002415(k+3), read by rows.

36, 120, 120, 300, 400, 300, 630, 1000, 1000, 630, 1176, 2100, 2500, 2100, 1176, 2016, 3920, 5250, 5250, 3920, 2016, 3240, 6720, 9800, 11025, 9800, 6720, 3240, 4950, 10800, 16800, 20580, 20580, 16800, 10800, 4950, 7260, 16500, 27000, 35280, 38416, 35280, 27000, 16500, 7260

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Aug 25 2008

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle begins as:
    36;
   120,   120;
   300,   400,   300;
   630,  1000,  1000,   630;
  1176,  2100,  2500,  2100,  1176;
  2016,  3920,  5250,  5250,  3920,  2016;
  3240,  6720,  9800, 11025,  9800,  6720,  3240;
  4950, 10800, 16800, 20580, 20580, 16800, 10800,  4950;
  7260, 16500, 27000, 35280, 38416, 35280, 27000, 16500, 7260;
		

Links

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

Crossrefs

Cf. A002415, A117662.

Programs

Magma

[Binomial((n-k+3)^2,2)*Binomial((k+3)^2,2)/36: k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 31 2024

Mathematica

A129367[n_, k_]:= Binomial[(n-k+3)^2, 2]*Binomial[(k+3)^2, 2]/36;
Table[A129367[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A129367(n,k): return binomial((n-k+3)^2,2)*binomial((k+3)^2,2)/36
flatten([[A129367(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 31 2024

Formula

T(n,k) = A002415(n-k+3)*A002415(k+3), where A002415(n) = n^2*(n^2-1)/12.

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

Extensions

Edited by G. C. Greubel, Jan 31 2024