A131402 2*A007318 - (A046854 + A065941 - A000012).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 6, 7, 6, 1, 1, 7, 14, 14, 7, 1, 1, 9, 20, 33, 20, 9, 1, 1, 10, 31, 56, 56, 31, 10, 1, 1, 12, 40, 97, 111, 97, 40, 12, 1, 1, 13, 55, 142, 217, 217, 142, 55, 13, 1, 1, 15, 67, 213, 358, 463, 358, 213, 67, 15, 1, 1, 16, 86, 287, 590, 841, 841, 590, 287, 86, 16, 1

Offset: 0

Author

Gary W. Adamson, Jul 07 2007

Keywords

• nonn
• tabl
• easy

Comments

Row sums = A131403: (1, 2, 5, 10, 21, 44, 93, ...).

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  4,  1;
  1,  6,  7,  6,  1;
  1,  7, 14, 14,  7,  1;
  1,  9, 20, 33, 20,  9,  1;
  1, 10, 31, 56, 56, 31, 10,  1;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

Crossrefs

Row sums are A131403.

Cf. A046854, A065941.

Programs

PARI

T(n,k) = if(k <= n, 2*binomial(n, k) + 1 - binomial((n + k)\2, k) - binomial(n-(k+1)\2, k\2), 0) \\ Andrew Howroyd, Aug 09 2018

Formula

2*A007318 - (A046854 + A065941 - A000012) as infinite lower triangular matrices.

T(n,k) = 2*binomial(n, k) + 1 - binomial(floor((n + k)/2), k) - binomial(n-floor((k+1)/2), floor(k/2)). - Andrew Howroyd, Aug 09 2018

Extensions

Missing terms inserted and a(55) and beyond from Andrew Howroyd, Aug 09 2018