

A129503


Pascal's FredholmRueppel 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
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OFFSET

1,5


COMMENTS

First row of the array = the FredholmRueppel 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 nth row (n=0,1,2,...) = M^n * V, where M = A115361 as an infinite lower triangular matrix and V = the FredholmRueppel 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 * nth 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(nk+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



