A157654 Triangle T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2, read by rows.

1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 4, 0, 4, 1, 1, 6, 2, 2, 6, 1, 1, 8, 4, 0, 4, 8, 1, 1, 10, 6, 2, 2, 6, 10, 1, 1, 12, 8, 4, 0, 4, 8, 12, 1, 1, 14, 10, 6, 2, 2, 6, 10, 14, 1, 1, 16, 12, 8, 4, 0, 4, 8, 12, 16, 1

Offset: 0

Author

Roger L. Bagula, Mar 03 2009

Keywords

• nonn
• tabl

Comments

For the cases of m = 0, 1 the triangles reduce to T(n, k, m) = A103451(n, k). - G. C. Greubel, Dec 13 2021

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  0,  1;
  1,  2,  2,  1;
  1,  4,  0,  4,  1;
  1,  6,  2,  2,  6,  1;
  1,  8,  4,  0,  4,  8,  1;
  1, 10,  6,  2,  2,  6, 10,  1;
  1, 12,  8,  4,  0,  4,  8, 12,  1;
  1, 14, 10,  6,  2,  2,  6, 10, 14,  1;
  1, 16, 12,  8,  4,  0,  4,  8, 12, 16, 1;
		

Links

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

Crossrefs

Cf. A000007, A103451, A244800.

Programs

Magma

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

Mathematica

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

Sage

def A157684(n,k,q): return 1 if (k==0 or k==n) else q*abs((n-k)^(q-1) - k^(q-1))
flatten([[A157684(n,k,2) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Dec 13 2021

Formula

T(n, k, m) = 1 if k = 0 or k = n, otherwise m*abs( (n-k)^(m-1) - k^(m-1) ), with m = 2.

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

Sum_{k=0..n} T(n, k, 2) = (-1)*[n==0] + A244800(n-1).

T(2*n, n, 2) = A000007(n). (End)

Extensions

Edited by G. C. Greubel, Dec 13 2021