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A125101
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T(n,k) = k*binomial(n-1,k-1) + Fibonacci(k)*binomial(n-1,k) (1 <= k <= n).
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0
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1, 2, 2, 3, 5, 3, 4, 9, 11, 4, 5, 14, 26, 19, 5, 6, 20, 50, 55, 30, 6, 7, 27, 85, 125, 105, 44, 7, 8, 35, 133, 245, 280, 182, 62, 8, 9, 44, 196, 434, 630, 560, 300, 85, 9, 10, 54, 276, 714, 1260, 1428, 1056, 477, 115, 10, 11, 65, 375, 1110, 2310, 3192, 3030, 1905, 745, 155
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OFFSET
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1,2
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COMMENTS
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Row sums are s(n) = 1, 4, 11, 28, 69, 167, 400, ...
Binomial transform of the bidiagonal matrix with (1,2,3...) in the main diagonal and the Fibonacci numbers (1,1,2,3,5,8,...) in the subdiagonal.
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LINKS
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FORMULA
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EXAMPLE
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First few rows of the triangle:
1;
2, 2;
3, 5, 3;
4, 9, 11, 4;
5, 14, 26, 19, 5;
6, 20, 50, 55, 30, 6;
7, 27, 85, 125, 105, 44, 7;
8, 35, 133, 245, 280, 182, 62, 8;
...
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MAPLE
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with(combinat): T:=(n, k)->k*binomial(n-1, k-1)+fibonacci(k)*binomial(n-1, k): for n from 1 to 12 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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MATHEMATICA
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Flatten[Table[k Binomial[n-1, k-1]+Fibonacci[k]Binomial[n-1, k], {n, 15}, {k, n}]] (* Harvey P. Dale, Nov 03 2014 *)
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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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