A134081 Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.

1, 2, 1, 3, 5, 1, 4, 12, 8, 1, 5, 22, 26, 11, 1, 6, 35, 60, 45, 14, 1, 7, 51, 115, 125, 69, 17, 1, 8, 70, 196, 280, 224, 98, 20, 1, 9, 92, 308, 546, 574, 364, 132, 23, 1, 10, 117, 456, 966, 1260, 1050, 552, 171, 26, 1

Offset: 0

Author

Gary W. Adamson, Oct 07 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle are:
  1;
  2,  1;
  3,  5,   1;
  4, 12,   8,   1;
  5, 22,  26,  11,  1;
  6, 35,  60,  45, 14,  1;
  7, 51, 115, 125, 69, 17, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Crossrefs

Cf. A002064, A007318, A112295.

Columns: A000027, A000326, A002413, A051740, A051879.

Programs

Magma

A134081:= func< n,k | Binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1) >;
[A134081(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

Mathematica

T[n_, k_]:= Binomial[n, k]*((2*k+1)*(n-k) +k+1)/(k+1);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

Sage

def A134081(n,k): return binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1)
flatten([[A134081(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

Formula

Binomial transform of A112295(unsigned).

From G. C. Greubel, Feb 17 2021: (Start)

T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1).

Sum_{k=0..n} T(n, k) = 2^n *n + 1 = A002064(n). (End)