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A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

%I A114320 #21 Aug 17 2023 18:52:12
%S A114320 1,1,1,1,3,3,15,6,3,75,30,15,435,225,45,15,3045,1575,315,105,24465,
%T A114320 12180,3150,420,105,220185,109620,28350,3780,945,2200905,1100925,
%U A114320 274050,47250,4725,945,24209955,12110175,3014550,519750,51975,10395,290529855
%N A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.
%C A114320 Row n has 1+floor(n/2) terms. Row sums yield the factorials (A000142). Sum(k*T(n,k),k>0)=n!/2 for n>=2. - _Emeric Deutsch_, Feb 17 2006
%H A114320 Alois P. Heinz, <a href="/A114320/b114320.txt">Rows n = 0..200, flattened</a>
%F A114320 E.g.f.: exp((y-1)*x^2/2)/(1-x). More generally, e.g.f. for number of permutations of n elements with k m-cycles is exp((y-1)*x^m/m)/(1-x).
%F A114320 T(n,k) = n!/(2^k*k!) * Sum_{j=0..floor(n/2)-k} (-1/2)^j/j!. - _Alois P. Heinz_, Nov 30 2011
%e A114320 T(3,1) = 3 because we have (1)(23), (12)(3) and (13)(2).
%e A114320 Triangle begins:
%e A114320     1;
%e A114320     1;
%e A114320     1,   1;
%e A114320     3,   3;
%e A114320    15,   6,   3;
%e A114320    75,  30,  15;
%e A114320   435, 225,  45,  15;
%e A114320   ...
%p A114320 G:= exp((y-1)*x^2/2)/(1-x): Gser:= simplify(series(G,x=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:= n!*coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(y*P[n], y^j), j=1..1+floor(n/2)) od;  # yields sequence in triangular form - _Emeric Deutsch_, Feb 17 2006
%t A114320 d = Exp[-x^2/2!]/(1 - x);f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Transpose[Table[Range[0, 10]!CoefficientList[Series[x^(2 k)/(2^k k!) d, {x, 0, 10}], x], {k, 0, 5}]]]]  (* _Geoffrey Critzer_, Nov 29 2011 *)
%Y A114320 Cf. A008290, A000266, A000090, A088436, A000138, A060725, A060726, A086659, A105422, A105114.
%K A114320 easy,nonn,tabf
%O A114320 0,5
%A A114320 _Vladeta Jovovic_, Feb 05 2006
%E A114320 More terms from _Emeric Deutsch_, Feb 17 2006