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A320508
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T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.
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1
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1, 1, -1, 1, 0, 1, 1, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 1, 0, 0, -1, 1, 4, 3, 0, 0, 0, 1, 1, 5, 6, 1, 0, 0, 0, -1, 1, 6, 10, 4, 0, 0, 0, 0, 1, 1, 7, 15, 10, 1, 0, 0, 0, 0, -1, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, -1, 1, 10, 36
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OFFSET
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0,12
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COMMENTS
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The n-th row consists of the coefficients in the expansion of (-x)^n + (((1 + sqrt(1 + 4*x))/2)^n -((1 - sqrt(1 + 4*x))/2)^n )/sqrt(1 + 4*x).
The coefficients in the expansion of Sum_{j=0..floor((n - 1)/2)} T(n,k)*x^(n - 2*j - 1) yield the n-th row in A168561, the coefficients of the n-th Fibonacci polynomial.
Row n sums up to Fibonacci(n) + (-1)^n (A008346).
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LINKS
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FORMULA
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G.f.: 1/((1 + x*y)*(1 - y - x*y^2)).
E.g.f.: exp(-x*y) + (exp(y*(1 + sqrt(1 + 4*x))/2) - exp(y*(1 - sqrt(1 + 4*x))/2))/sqrt(1 + 4*x).
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EXAMPLE
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Triangle begins:
1;
1, -1;
1, 0, 1;
1, 1, 0, -1;
1, 2, 0, 0, 1;
1, 3, 1, 0, 0, -1;
1, 4, 3, 0, 0, 0, 1;
1, 5, 6, 1, 0, 0, 0, -1;
1, 6, 10, 4, 0, 0, 0, 0, 1;
1, 7, 15, 10, 1, 0, 0, 0, 0, -1;
1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1;
1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, -1;
...
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MATHEMATICA
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Table[Table[Binomial[n - k - 1, k], {k, 0, n}], {n, 0, 12}]//Flatten
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PROG
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(Maxima) create_list(binomial(n - k - 1, k), n, 0, 12, k, 0, n);
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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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