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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).

%I A145880 #16 Jul 06 2023 06:31:19
%S A145880 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,
%T A145880 3816,1681,120,1,0,246,6210,26916,26916,6210,246,0,1,502,21433,160054,
%U A145880 303505,160054,21433,502,1,0,1012,70774,852346,2747008,2747008,852346,70774
%N 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).
%C A145880 Row n has n-1 entries (n>=2).
%C A145880 Sum of entries in row n = A000387(n).
%C A145880 Sum_{k=1..n-1} k*T(n,k) = A145886(n) (n>=2).
%H A145880 R. Mantaci and F. Rakotondrajao, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00531-6">Exceedingly deranging!</a>, Advances in Appl. Math., 30 (2003), 177-188.
%F A145880 E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) + (t*exp(-z)-exp(-tz))/(1-t))/2.
%e A145880 T(4,2)=4 because the odd derangements of {1,2,3,4} with 2 excedances are 3142, 4312, 2413 and 3421.
%e A145880 Triangle starts:
%e A145880   0;
%e A145880   1;
%e A145880   0,  0;
%e A145880   1,  4,  1;
%e A145880   0, 10, 10,  0;
%e A145880   1, 26, 81, 26,  1;
%p A145880 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
%Y A145880 Cf. A000387, A145886, A145881, A145887.
%K A145880 nonn,tabf
%O A145880 1,6
%A A145880 _Emeric Deutsch_, Nov 06 2008
%E A145880 Formula corrected by _N. J. A. Sloane_, Jul 20 2017 at the request of the author.