A129503 Pascal's Fredholm-Rueppel triangle.

1, 1, 1, 1, 2, 0, 1, 3, 0, 1, 1, 4, 0, 3, 0, 1, 5, 0, 6, 0, 0, 1, 6, 0, 10, 0, 0, 0, 1, 7, 0, 15, 0, 0, 0, 1, 1, 8, 0, 21, 0, 0, 0, 4, 0, 1, 9, 0, 28, 0, 0, 0, 10, 0, 0, 1, 10, 0, 36, 0, 0, 0, 20, 0, 0, 0, 1, 11, 0, 45, 0, 0, 0, 35, 0, 0, 0, 0, 1, 12, 0, 55, 0, 0, 0, 56, 0, 0, 0, 0, 0

Offset: 1

Author

Gary W. Adamson, Apr 18 2007

Keywords

• nonn
• tabl

Comments

First row of the array = the Fredholm-Rueppel sequence (A036987); which becomes the right border of the triangle. Second row of the array (1, 2, 0, 3, 0, 0, 0, 4, ...) = A104117. Third row of the array (1, 3, 0, 6, 0, 0, 0, 10, ...) = A129502. Row sums of triangle A129503 = A129504: (1, 2, 3, 5, 8, 12, 17, 24, 34, ...).

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  2,  0;
  1,  3,  0,  1;
  1,  4,  0,  3,  0;
  1,  5,  0,  6,  0,  0;
  1,  6,  0, 10,  0,  0,  0;
  1,  7,  0, 15,  0,  0,  0,  1;
  1,  8,  0, 21,  0,  0,  0,  4,  0;
  1,  9,  0, 28,  0,  0,  0, 10,  0,  0;
  1, 10,  0, 36,  0,  0,  0, 20,  0,  0,  0;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275

Crossrefs

Row sums are A129504.

Cf. A036987, A115361, A104117, A129502.

Programs

PARI

T(n,k)=my(e=valuation(k,2)); if(k==2^e, binomial(n-k+e, e)) \\ Andrew Howroyd, Aug 09 2018

Formula

Antidiagonals of an array in which n-th row (n=0,1,2,...) = M^n * V, where M = A115361 as an infinite lower triangular matrix and V = the Fredholm-Rueppel sequence A036987 as a vector: [1, 1, 0, 1, 0, 0, 0, 1, ...]. The array = 1, 1, 0, 1, 0, 0, 0, 1, 0, ... 1, 2, 0, 3, 0, 0, 0, 4, 0, ... 1, 3, 0, 6, 0, 0, 0, 10, 0, ... 1, 4, 0, 10, 0, 0, 0, 20, 0, ... (n+1)-th row can be generated from A115361 * n-th row.

T(n, 2^e) = binomial(n + e - 2^e, e), T(n, k) = 0 otherwise. - Andrew Howroyd, Aug 09 2018

Extensions

a(53) corrected and terms a(67) and beyond from Andrew Howroyd, Aug 09 2018