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A060523
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Triangle T(n,k) = number of degree-n permutations with k even cycles, k=0..n.
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4
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1, 1, 0, 1, 1, 0, 3, 3, 0, 0, 9, 12, 3, 0, 0, 45, 60, 15, 0, 0, 0, 225, 345, 135, 15, 0, 0, 0, 1575, 2415, 945, 105, 0, 0, 0, 0, 11025, 18480, 9030, 1680, 105, 0, 0, 0, 0, 99225, 166320, 81270, 15120, 945, 0, 0, 0, 0, 0, 893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0
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OFFSET
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0,7
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REFERENCES
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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, p. 189, Exercise 3.3.13.
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LINKS
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FORMULA
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E.g.f.: (1+x)^((1-y)/2)/(1-x)^((1+y)/2).
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EXAMPLE
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Triangle T(n,k) begins:
1;
1, 0;
1, 1, 0;
3, 3, 0, 0;
9, 12, 3, 0, 0;
45, 60, 15, 0, 0, 0;
225, 345, 135, 15, 0, 0, 0;
1575, 2415, 945, 105, 0, 0, 0, 0;
11025, 18480, 9030, 1680, 105, 0, 0, 0, 0;
99225, 166320, 81270, 15120, 945, 0, 0, 0, 0, 0;
893025, 1596105, 897750, 217350, 23625, 945, 0, 0, 0, 0, 0;
...
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MAPLE
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with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
`if`(irem(i, 2)=0, x^j, 1), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
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MATHEMATICA
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nn = 6; Range[0, nn]! CoefficientList[
Series[(1 - x^2)^(-y/2) ((1 + x)/(1 - x))^(1/2), {x, 0, nn}], {x, y}] // Grid (* Geoffrey Critzer, Aug 27 2012 *)
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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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