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A083857
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Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.
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4
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0, 0, 1, 0, 1, 3, 0, 1, 3, 7, 0, 1, 3, 8, 15, 0, 1, 3, 9, 21, 31, 0, 1, 3, 10, 27, 55, 63, 0, 1, 3, 11, 33, 81, 144, 127, 0, 1, 3, 12, 39, 109, 243, 377, 255, 0, 1, 3, 13, 45, 139, 360, 729, 987, 511, 0, 1, 3, 14, 51, 171, 495, 1189, 2187, 2584, 1023, 0, 1, 3, 15, 57, 205, 648
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OFFSET
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0,6
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COMMENTS
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Row n >= 0 of the array gives the solution to the recurrence b(k) = 3*b(k-1) + (n-2) * a(k-2) for k >= 2 with a(0) = 0 and a(1) = 1. These are the binomial transforms of the rows of the generalized Fibonacci numbers A083856.
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LINKS
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FORMULA
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T(n, k) = ((3 + sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) - ((3 - sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) for n, k >= 0.
O.g.f. of row n >= 0: -x/(-1 + 3*x + (n-2)*x^2) . - R. J. Mathar, Nov 23 2007
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EXAMPLE
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Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
0, 1, 3, 7, 15, 31, 63, 127, 255, ...
0, 1, 3, 8, 21, 55, 144, 377, 987, ...
0, 1, 3, 9, 27, 81, 243, 729, 2187, ...
0, 1, 3, 10, 33, 109, 360, 1189, 3927, ...
0, 1, 3, 11, 39, 139, 495, 1763, 6279, ...
0, 1, 3, 12, 45, 171, 648, 2457, 9315, ...
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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