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

0,2

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

Cf. A002064, A007318, A112295.

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

Sequence in context: A210233 A347667 A297582 * A134247 A210225 A180906

Adjacent sequences: A134078 A134079 A134080 * A134082 A134083 A134084

Keyword

nonn,tabl

Author

Gary W. Adamson, Oct 07 2007

Status

approved