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A145881
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Triangle read by rows: T(n,k) is the number of even permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).
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3
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0, 0, 1, 1, 0, 3, 0, 1, 11, 11, 1, 0, 25, 80, 25, 0, 1, 57, 407, 407, 57, 1, 0, 119, 1680, 3815, 1680, 119, 0, 1, 247, 6211, 26917, 26917, 6211, 247, 1, 0, 501, 21432, 160053, 303504, 160053, 21432, 501, 0, 1, 1013, 70775, 852347, 2747009, 2747009, 852347, 70775
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OFFSET
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1,6
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COMMENTS
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Row n has n-1 entries (n>=2).
Sum of entries in row n = A000321(n).
Sum_{k=1..n-1} k*T(n,k) = A145887(n) (n>=2).
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LINKS
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FORMULA
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E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) - (t*exp(-z)-exp(-tz))/(1-t))/2.
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EXAMPLE
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T(4,2)=3 because the even derangements of {1,2,3,4} are 3412, 2143 and 4321.
Triangle starts:
0;
0;
1, 1;
0, 3, 0;
1, 11, 11, 1;
0, 25, 80, 25, 0;
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MAPLE
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G:=((1-t)*exp(-t*z)/(1-t*exp((1-t)*z))-(t*exp(-z)-exp(-t*z))/(1-t))*1/2: Gser:=simplify(series(G, z=0, 15)): for n to 11 do P[n]:=sort(expand(factorial(n)*coeff(Gser, z, n))) end do: 0; for n to 11 do seq(coeff(P[n], t, j), j=1..n-1) end do; # yields sequence in triangular form
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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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EXTENSIONS
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STATUS
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approved
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