A155494 Triangle T(n, k) = (k+1)*(n-k+1)*binomial(n,k) with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 8, 1, 1, 18, 18, 1, 1, 32, 54, 32, 1, 1, 50, 120, 120, 50, 1, 1, 72, 225, 320, 225, 72, 1, 1, 98, 378, 700, 700, 378, 98, 1, 1, 128, 588, 1344, 1750, 1344, 588, 128, 1, 1, 162, 864, 2352, 3780, 3780, 2352, 864, 162, 1, 1, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 1

Offset

0,5

Links

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

Formula

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

Sum_{k=0..n} T(n, k) = 2^(n-2)*(n^2 +3*n +4) -2*n. - G. C. Greubel, May 27 2021

Example

Triangle begins as:
  1;
  1,   1;
  1,   8,    1;
  1,  18,   18,    1;
  1,  32,   54,   32,    1;
  1,  50,  120,  120,   50,    1;
  1,  72,  225,  320,  225,   72,    1;
  1,  98,  378,  700,  700,  378,   98,    1;
  1, 128,  588, 1344, 1750, 1344,  588,  128,    1;
  1, 162,  864, 2352, 3780, 3780, 2352,  864,  162,   1;
  1, 200, 1215, 3840, 7350, 9072, 7350, 3840, 1215, 200, 1;

Mathematica

T[n_, k_]:= If[k*(n-k)==0, 1, (k+1)*(n-k+1)*Binomial[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 27 2021 *)

Prog

(Magma)
A155494:= func< n, k | k eq 0 or k eq n select 1 else (k+1)*(n-k+1)*Binomial(n, k) >;
[A155494(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 27 2021
(Sage)
def A155494(n, k): return 1 if (k==0 or k==n) else (k+1)*(n-k+1)*binomial(n, k)
flatten([[A155494(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 27 2021

Crossrefs

Sequence in context: A174125 A051425 A051469 * A154227 A166340 A157178

Adjacent sequences: A155491 A155492 A155493 * A155495 A155496 A155497

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Jan 23 2009

Extensions

Edited by G. C. Greubel, May 27 2021

Status

approved