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A144389
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Triangle T(n,k) = n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), T(0,0) = 1, read by rows, 0 <= k <= n.
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1
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-1, 2, -1, 1, 4, -1, 4, 3, 6, -1, 3, 16, 6, 8, -1, 6, 15, 40, 10, 10, -1, 5, 36, 45, 80, 15, 12, -1, 8, 35, 126, 105, 140, 21, 14, -1, 7, 64, 140, 336, 210, 224, 28, 16, -1, 10, 63, 288, 420, 756, 378, 336, 36, 18, -1, 9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1
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text;
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OFFSET
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0,2
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LINKS
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FORMULA
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T(n,k) = [x^k] (n*(x + 1)^(n - 1) - (x - 1)^n).
Sum_{k=0..n} T(n,k) = A001787(n), n >= 1.
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EXAMPLE
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Triangle begins:
-1;
2, -1;
1, 4, -1;
4, 3, 6, -1;
3, 16, 6, 8, -1;
6, 15, 40, 10, 10, -1;
5, 36, 45, 80, 15, 12, -1;
8, 35, 126, 105, 140, 21, 14, -1;
7, 64, 140, 336, 210, 224, 28, 16, -1;
10, 63, 288, 420, 756, 378, 336, 36, 18, -1;
9, 100, 315, 960, 1050, 1512, 630, 480, 45, 20, -1;
...
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MATHEMATICA
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p[x_, n_] = -(x - 1)^n + n*(x + 1)^(n - 1);
Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten
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PROG
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(Maxima) create_list(n*binomial(n - 1, k) - (-1)^(n - k)*binomial(n, k), n , 0, 15, k, 0, n); /* Franck Maminirina Ramaharo, Jan 25 2019 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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