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

0,8

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

Links

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

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)

Example

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;

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

Prog

(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
(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

Crossrefs

Cf. A000007, A103451, A244800.

Sequence in context: A342061 A045995 A360625 * A357437 A078692 A273432

Adjacent sequences: A157651 A157652 A157653 * A157655 A157656 A157657

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 03 2009

Extensions

Edited by G. C. Greubel, Dec 13 2021

Status

approved