OFFSET

1,5

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, ...).

LINKS

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

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

EXAMPLE

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;

...

PROG

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

CROSSREFS

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Apr 18 2007

EXTENSIONS

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

STATUS

approved