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A145880
Triangle read by rows: T(n,k) is the number of odd permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).
3
0, 1, 0, 0, 1, 4, 1, 0, 10, 10, 0, 1, 26, 81, 26, 1, 0, 56, 406, 406, 56, 0, 1, 120, 1681, 3816, 1681, 120, 1, 0, 246, 6210, 26916, 26916, 6210, 246, 0, 1, 502, 21433, 160054, 303505, 160054, 21433, 502, 1, 0, 1012, 70774, 852346, 2747008, 2747008, 852346, 70774
OFFSET
1,6
COMMENTS
Row n has n-1 entries (n>=2).
Sum of entries in row n = A000387(n).
Sum_{k=1..n-1} k*T(n,k) = A145886(n) (n>=2).
LINKS
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188.
FORMULA
E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) + (t*exp(-z)-exp(-tz))/(1-t))/2.
EXAMPLE
T(4,2)=4 because the odd derangements of {1,2,3,4} with 2 excedances are 3142, 4312, 2413 and 3421.
Triangle starts:
0;
1;
0, 0;
1, 4, 1;
0, 10, 10, 0;
1, 26, 81, 26, 1;
MAPLE
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 06 2008
EXTENSIONS
Formula corrected by N. J. A. Sloane, Jul 20 2017 at the request of the author.
STATUS
approved