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A123218
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Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2)* binomial(n - floor((k + 1)/2), floor(k/2)).
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2
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1, 1, -2, -1, 1, -2, -6, 2, 1, 1, -2, -11, 12, 11, -2, -1, 1, -2, -16, 22, 46, -22, -16, 2, 1, 1, -2, -21, 32, 106, -92, -106, 32, 21, -2, -1, 1, -2, -26, 42, 191, -212, -396, 212, 191, -42, -26, 2, 1, 1, -2, -31, 52, 301, -382, -1011, 792, 1011, -382, -301, 52, 31, -2, -1
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OFFSET
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1,3
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LINKS
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FORMULA
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Let f(n, k) = (-1)^floor((k + 1)/2)*binomial(n - floor((k + 1)/2), floor(k/2)) then the polynomials P(n, k, x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k) for an irregular triangle of coefficients.
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EXAMPLE
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Triangle begins with:
1;
1, -2, -1;
1, -2, -6, 2, 1;
1, -2, -11, 12, 11, -2, -1;
1, -2, -16, 22, 46, -22, -16, 2, 1;
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MATHEMATICA
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f[n_, k_]:= (-1)^Floor[(k+1)/2]*Binomial[n -Floor[(k+1)/2], Floor[k/2]]; Table[CoefficientList[Sum[f[n, k]*(2*x)^k*(1-x^2)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten
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CROSSREFS
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KEYWORD
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tabf,sign
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AUTHOR
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STATUS
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approved
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