A131087 Triangle read by rows: T(n,k) = 2*binomial(n,k) - (1 + (-1)^(n-k))/2 (0 <= k <= n).

1, 2, 1, 1, 4, 1, 2, 5, 6, 1, 1, 8, 11, 8, 1, 2, 9, 20, 19, 10, 1, 1, 12, 29, 40, 29, 12, 1, 2, 13, 42, 69, 70, 41, 14, 1, 1, 16, 55, 112, 139, 112, 55, 16, 1, 2, 17, 72, 167, 252, 251, 168, 71, 18, 1, 1, 20, 89, 240, 419, 504, 419, 240, 89, 20, 1, 2, 21, 110, 329, 660, 923, 924

Offset: 0

Author

Gary W. Adamson, Jun 14 2007

Keywords

• nonn
• tabl

Comments

Row sums = A084174: (1, 3, 6, 14, 29, ...).

2*A007318 - A128174 as infinite lower triangular matrices. - Emeric Deutsch, Jun 21 2007

Examples

			First few rows of the triangle:
  1;
  2,  1;
  1,  4,  1;
  2,  5,  6,  1;
  1,  8, 11,  8,  1;
  2,  9, 20, 19, 10,  1;
  1, 12, 29, 40, 29, 12,  1;
  ...
		

Crossrefs

Cf. A128174, A084174.

Programs

Maple

T := proc (n, k) options operator, arrow; 2*binomial(n, k)-1/2-(1/2)*(-1)^(n-k) end proc; for n from 0 to 11 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jun 21 2007

Formula

G.f.: G(t,z) = (1 + z - tz - 2z^2 + 2tz^3)/((1-z^2)*(1-tz)*(1-z-tz)). - Emeric Deutsch, Jun 21 2007

Extensions

More terms from Emeric Deutsch, Jun 21 2007