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%I #17 Feb 09 2022 09:46:43
%S 1,0,0,1,2,0,6,0,3,24,20,0,160,90,0,15,1140,504,210,0,8988,4480,1260,
%T 0,105,80864,41040,9072,2520,0,809856,404460,100800,18900,0,945,
%U 8907480,4447520,1128600,166320,34650,0,106877320,53450496,13347180,2217600
%N 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)).
%C Row n has 1 + floor(n/2) entries.
%C Sum of entries in row n = A000166(n) (the derangement numbers).
%C T(n,0) = A038205(n).
%C Sum_{k>=0} k*T(n,k) = A000387(n).
%H Alois P. Heinz, <a href="/A162974/b162974.txt">Rows n = 0..200, flattened</a>
%F E.g.f.: G(t,z) = exp(z(tz-z-2)/2)/(1-z).
%e T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
%e Triangle starts:
%e 1;
%e 0;
%e 0, 1;
%e 2, 0;
%e 6, 0, 3;
%e 24, 20, 0;
%e 160, 90, 0, 15;
%e ...
%p 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;
%p # second Maple program:
%p b:= proc(n) option remember; expand(`if`(n=0, 1, add((j-1)!*
%p `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=2..n)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
%p seq(T(n), n=0..14); # _Alois P. Heinz_, Jan 27 2022
%Y Cf. A000166, A000387, A038205, A114320.
%Y T(2n,n) gives A001147.
%Y T(2n+3,n) gives A000906(n) = 2*A000457(n).
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Jul 22 2009
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