A157636 Triangle read by rows: T(n, k) = 1 if k=0 or k=n, otherwise = n*k*(n-k)/2.

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 8, 6, 1, 1, 10, 15, 15, 10, 1, 1, 15, 24, 27, 24, 15, 1, 1, 21, 35, 42, 42, 35, 21, 1, 1, 28, 48, 60, 64, 60, 48, 28, 1, 1, 36, 63, 81, 90, 90, 81, 63, 36, 1, 1, 45, 80, 105, 120, 125, 120, 105, 80, 45, 1

Offset: 0

Author

Roger L. Bagula, Mar 03 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,  1;
  1,  3,  3,   1;
  1,  6,  8,   6,   1;
  1, 10, 15,  15,  10,   1;
  1, 15, 24,  27,  24,  15,   1;
  1, 21, 35,  42,  42,  35,  21,   1;
  1, 28, 48,  60,  64,  60,  48,  28,  1;
  1, 36, 63,  81,  90,  90,  81,  63, 36,  1;
  1, 45, 80, 105, 120, 125, 120, 105, 80, 45, 1;
		

Links

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

Crossrefs

Cf. A000578, A002415, A107985, A132411.

Programs

Magma

A157636:= func< n,k | k eq 0 or k eq n select 1 else n*k*(n-k)/2 >;
[A157636(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 13 2021

Mathematica

T[n_, k_] = If[n*k*(n-k)==0, 1, n*k*(n-k)/2];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten

Sage

def A157636(n,k): return 1 if (k==0 or k==n) else n*k*(n-k)/2
flatten([[A157636(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021

Formula

T(n, k) = 1 if k=0 or k=n, otherwise = n*k*(n-k)/2.

Sum_{k=0..n} T(n, k) = 2 + n^2*(n^2 - 1)/12 = 2 + A002415(n) if n>0.

From G. C. Greubel, Dec 13 2021: (Start)

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

T(n, 1) = [n<2] + binomial(n, 2).

T(n, 2) = A132411(n-1), for n >= 2.

T(2*n, n) = [n=0] + A000578(n). (End)