A125103 Triangle read by rows: T(n,k) = binomial(n,k) + 2^k*binomial(n,k+1) (0 <= k <= n).

1, 2, 1, 3, 4, 1, 4, 9, 7, 1, 5, 16, 22, 12, 1, 6, 25, 50, 50, 21, 1, 7, 36, 95, 140, 111, 38, 1, 8, 49, 161, 315, 371, 245, 71, 1, 9, 64, 252, 616, 966, 952, 540, 136, 1, 10, 81, 372, 1092, 2142, 2814, 2388, 1188, 265, 1, 11, 100, 525, 1800, 4242, 6972, 7890, 5880, 2605, 522, 1

Offset: 0

Author

Gary W. Adamson, Nov 20 2006

Keywords

• nonn
• tabl

Comments

Row sums = A094374: (1, 3, 8, 21, 56, ...).

Binomial transform of the infinite bidiagonal matrix with (1,1,1,...) in the main diagonal and (1,2,4,8,...) in the subdiagonal.

Examples

			First few rows of the triangle are
  1;
  2,   1;
  3,   4,   1;
  4,   9,   7,   1;
  5,  16,  22,  12,   1;
  6,  25,  50,  50,  21,   1;
  7,  36,  95, 140, 111,  38,  1;
  ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A094374.

Programs

Maple

T:=(n,k)->binomial(n,k)+2^k*binomial(n,k+1): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

Mathematica

Table[Binomial[n,k]+2^k Binomial[n,k+1],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Nov 30 2019 *)

PARI

T(n,k) = binomial(n,k) + 2^k*binomial(n,k+1);
matrix(11, 11, n, k, T(n-1,k-1)) \\ Michel Marcus, Nov 09 2019

Extensions

Edited by N. J. A. Sloane, Nov 29 2006