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A057094
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Coefficient triangle for certain polynomials (rising powers).
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1
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0, 0, -1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 2, -1, 0, 0, 0, -1, 3, -1, 0, 0, 0, 0, -3, 4, -1, 0, 0, 0, 0, 1, -6, 5, -1, 0, 0, 0, 0, 0, 4, -10, 6, -1, 0, 0, 0, 0, 0, -1, 10, -15, 7, -1, 0, 0, 0, 0, 0, 0, -5, 20, -21, 8, -1, 0, 0, 0, 0, 0, 0, 1, -15, 35, -28, 9, -1, 0, 0, 0, 0, 0, 0, 0, 6, -35, 56, -36, 10, -1, 0, 0, 0, 0, 0, 0, 0, -1, 21, -70, 84
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OFFSET
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0,14
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COMMENTS
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The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are negative scaled Chebyshev U-polynomials: p(n,x)= -U(n-1,sqrt(x)/2)*(sqrt(x))^(n+1), n >= 1. p(0,x)=0. p(n-1,1/x) appears in the n-th power of the g.f. of Catalan's numbers A000108, c(x): (c(x))^n = p(n-1,1/x)*1 -p(n,1/x)*x*c(x). Cf. Lang reference eqs.(1) and (2).
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LINKS
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FORMULA
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a(n, m)=0 if n<m; a(0, 0)=0; a(n, m)= ((-1)^(n-m+1))*binomial(m-1, n-m) if n >= 1 and n >= m >=floor(n/2)+1; else 0.
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EXAMPLE
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Triangle begins:
0;
0, -1;
0, 0, -1;
0, 0, 1, -1;
0, 0, 0, 2, -1;
0, 0, 0, -1, 3, -1;
...
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MATHEMATICA
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Prepend[CoefficientList[Table[I^n x^(n/2) Fibonacci[n - 1, -I Sqrt[x]], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Prepend[CoefficientList[Table[-x^(n/2) ChebyshevU[n - 2, Sqrt[x]/2], {n, 2, 14}], x], {0}] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
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PROG
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(PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, if ((n==0) || (k < n\2+1), v = 0, v = (-1)^(n-k+1)*binomial(k-1, n-k)); print1(v, ", "); ); print(); ); } \\ Michel Marcus, Jan 14 2016
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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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