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A123199
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Irregular triangle read by rows: row n is the expansion of (1 + 2*x - x^2)^n.
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13
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1, 1, 2, -1, 1, 4, 2, -4, 1, 1, 6, 9, -4, -9, 6, -1, 1, 8, 20, 8, -26, -8, 20, -8, 1, 1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1, 1, 12, 54, 100, 15, -168, -76, 168, 15, -100, 54, -12, 1, 1, 14, 77, 196, 161, -238, -427, 184, 427, -238, -161, 196, -77, 14
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OFFSET
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0,3
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COMMENTS
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The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A007318(n, j)*(2*x)^j*(1 - x^2)^(n-j).
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REFERENCES
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Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, p. 164, figure 26.1.
Henry McKean and Victor Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic, Cambridge University Press, 1997, p. 106, figure 2.22.
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LINKS
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FORMULA
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Row n is made of coefficients of: (1 + 2*x - x^2)^n. - Thomas Baruchel, Jan 15 2015
G.f.: 1/(1 - (1 + 2*x - x^2)*y).
E.g.f.: exp((1 + 2*x - x^2)*y).
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EXAMPLE
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Triangle begins:
1;
1, 2, -1;
1, 4, 2, -4, 1;
1, 6, 9, -4, -9, 6, -1;
1, 8, 20, 8, -26, -8, 20, -8, 1;
1, 10, 35, 40, -30, -68, 30, 40, -35, 10, -1;
...
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MATHEMATICA
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Table[CoefficientList[(-x^2 + 2*x + 1)^n, x], {n, 0, 10}]//Flatten
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PROG
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(Sage)
def T(n): return ( (1+2*x-x^2)^n ).full_simplify().coefficients(sparse=False)
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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