A157528 Triangle read by rows: T(n, k) = 2*k*(n - k) with T(n, 0) = T(n, n) = 1.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 8, 6, 1, 1, 8, 12, 12, 8, 1, 1, 10, 16, 18, 16, 10, 1, 1, 12, 20, 24, 24, 20, 12, 1, 1, 14, 24, 30, 32, 30, 24, 14, 1, 1, 16, 28, 36, 40, 40, 36, 28, 16, 1, 1, 18, 32, 42, 48, 50, 48, 42, 32, 18, 1

Offset

0,5

Links

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

Formula

T(n, k) = 2*k*(n - k) with T(n, 0) = T(n, n) = 1.

Sum_{k=0..n} T(n, k) = (n+2)*(n^2 - 2*n + 3)/3 - [n=0].

Example

Triangle begins
  1;
  1,  1;
  1,  2,  1;
  1,  4,  4,  1;
  1,  6,  8,  6,  1;
  1,  8, 12, 12,  8,  1;
  1, 10, 16, 18, 16, 10,  1;
  1, 12, 20, 24, 24, 20, 12,  1;
  1, 14, 24, 30, 32, 30, 24, 14,  1;
  1, 16, 28, 36, 40, 40, 36, 28, 16,  1;
  1, 18, 32, 42, 48, 50, 48, 42, 32, 18,  1;

Maple

A157528 := proc(n, k) if k=0 or k=n then 1; else 2*k*(n-k) ; end if; end proc: # R. J. Mathar, Sep 14 2011

Mathematica

T[n_, k_] = With[{m=2}, If[k*(n-k)==0, 1, n^m - (k^m + (n-k)^m)]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Magma) [k eq 0 or k eq n select 1 else 2*k*(n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 14 2022
(Sage) flatten([[1 if (k==0 or k==n) else 2*k*(n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 14 2022

Crossrefs

Sequence in context: A147532 A283796 A156580 * A132731 A128966 A055907

Adjacent sequences: A157525 A157526 A157527 * A157529 A157530 A157531

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Mar 02 2009

Status

approved