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A129503
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Pascal's Fredholm-Rueppel triangle.
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2
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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
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1,5
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COMMENTS
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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, ...).
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LINKS
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FORMULA
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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
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EXAMPLE
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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;
...
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PROG
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(PARI) T(n, k)=my(e=valuation(k, 2)); if(k==2^e, binomial(n-k+e, e)) \\ Andrew Howroyd, Aug 09 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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a(53) corrected and terms a(67) and beyond from Andrew Howroyd, Aug 09 2018
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STATUS
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approved
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