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A178111
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Number triangle T(n,k)=(-1)^((n-k)/2)*C(floor(n/2),floor(k/2))*(1+(-1)^(n-k))/2.
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2
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1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 1, 0, -2, 0, 1, 0, 1, 0, -2, 0, 1, -1, 0, 3, 0, -3, 0, 1, 0, -1, 0, 3, 0, -3, 0, 1, 1, 0, -4, 0, 6, 0, -4, 0, 1, 0, 1, 0, -4, 0, 6, 0, -4, 0, 1, -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1, 0, -1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1, 1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1
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OFFSET
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0,13
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COMMENTS
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Coefficient array of orthogonal polynomials P(n,x)=xP(n-1,x)-((1+(-1)^n)/2)*P(n-2,x), P(0,x)=1,P(1,x)=x.
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LINKS
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EXAMPLE
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Triangle begins
1,
0, 1,
-1, 0, 1,
0, -1, 0, 1,
1, 0, -2, 0, 1,
0, 1, 0, -2, 0, 1,
-1, 0, 3, 0, -3, 0, 1,
0, -1, 0, 3, 0, -3, 0, 1,
1, 0, -4, 0, 6, 0, -4, 0, 1,
0, 1, 0, -4, 0, 6, 0, -4, 0, 1,
-1, 0, 5, 0, -10, 0, 10, 0, -5, 0, 1
Production matrix is
0, 1,
-1, 0, 1,
0, 0, 0, 1,
0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, -1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Production matrix of inverse is
0, 1,
1, 0, 1,
0, 0, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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MAPLE
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P := (n, x) -> `if`(n < 2, x^n, x*P(n-1, x) - ((1+(-1)^n)/2)*P(n-2, x)):
ListTools:-Flatten([seq(PolynomialTools:-CoefficientList(P(n, x), x), n=0..12)]);
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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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