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A129503 Pascal's Fredholm-Rueppel triangle. 2
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 (list; table; graph; refs; listen; history; text; internal format)
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 are:

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

Row sums are A129504.

Cf. A036987, A115361, A104117, A129502.

Sequence in context: A278045 A096335 A191910 * A225682 A144185 A143987

Adjacent sequences:  A129500 A129501 A129502 * A129504 A129505 A129506

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

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Last modified July 21 21:14 EDT 2019. Contains 325199 sequences. (Running on oeis4.)