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A162974
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Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having k cycles of length 2 (0 <= k <= floor(n/2)).
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2
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1, 0, 0, 1, 2, 0, 6, 0, 3, 24, 20, 0, 160, 90, 0, 15, 1140, 504, 210, 0, 8988, 4480, 1260, 0, 105, 80864, 41040, 9072, 2520, 0, 809856, 404460, 100800, 18900, 0, 945, 8907480, 4447520, 1128600, 166320, 34650, 0, 106877320, 53450496, 13347180, 2217600
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OFFSET
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0,5
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COMMENTS
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Row n has 1 + floor(n/2) entries.
Sum of entries in row n = A000166(n) (the derangement numbers).
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LINKS
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FORMULA
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E.g.f.: G(t,z) = exp(z(tz-z-2)/2)/(1-z).
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EXAMPLE
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T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
Triangle starts:
1;
0;
0, 1;
2, 0;
6, 0, 3;
24, 20, 0;
160, 90, 0, 15;
...
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MAPLE
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G := exp((1/2)*z*(t*z-z-2))/(1-z): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do;
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add((j-1)!*
`if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=2..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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