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A141664
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An irregular triangular sequence formed by partition-like complex polynomials.
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1
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1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, -1, -1, 0, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, -2, -2, -2, -1, -1, 1, 1, 1, 1, 1, 0, 1, 0, 0, -1, -1, -2, -2, -3, -2, -2, 0, 0, 2, 2, 3, 2, 2, 1, 1, 0, 0, -1, 1, 0, 0, -1, -1, -2, -2, -3, -3, -3, -1, -1, 1, 2, 4, 4, 5, 4, 4, 3, 2, 0, 0, -1, -1, -1, -1, -1, 0, 1, 0, 0, -1, -1, -2, -2, -3, -3, -4, -2, -2, 0
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OFFSET
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0,20
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COMMENTS
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Row sums are: {1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, ...}.
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LINKS
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FORMULA
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Let p(x,n) = Product_{j=1..n} (1 + i*x^j), p(x,0)=1, with i being the imaginary unit, then the n-th row is the real part of the coefficients of p(x,n).
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EXAMPLE
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Irregular triangle begins as:
1.
1, 0.
1, 0, 0, -1.
1, 0, 0, -1, -1, -1, 0.
1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1.
1, 0, 0, -1, -1, -2, -2, -2, -1, -1, 1, 1, 1, 1, 1, 0. ...
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MATHEMATICA
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p[x_, n_]:= If[n == 0, 1, Product[(1 + I*x^i), {i, 1, n}]];
Table[Expand[p[x, n]], {n, 0, 10}];
Table[Re[CoefficientList[p[x, n], x]], {n, 0, 10}]//Flatten
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PROG
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(PARI) row(n) = if (n==0, 1, apply(x->real(x), Vecrev(prod(j=1, n, (1 + I*x^j))))); \\ Michel Marcus, Apr 02 2019
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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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