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A157018 Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0

%I A157018 #15 Nov 05 2016 08:13:03
%S A157018 1,3,6,6,10,30,15,90,90,21,210,630,28,420,2520,2520,36,756,7560,22680,
%T A157018 45,1260,18900,113400,113400,55,1980,41580,415800,1247400,66,2970,
%U A157018 83160,1247400,7484400,7484400,78,4290,154440,3243240,32432400,97297200
%N A157018 Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0<k<=floor(n/2).
%C A157018 T(n,k) is also the number of involutions (unary operators) on S_n, i.e., endomorphisms U with 2k non-invariant elements such that U^2 is the identity mapping. The extension to n=1 is a(1)=0. - _Stanislav Sykora_, Nov 03 2016
%H A157018 Stanislav Sykora, <a href="/A157018/b157018.txt">Table of n, a(n) for n = 2..2501</a>
%H A157018 Wikipedia, <a href="http://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>.
%F A157018 E.g.f.: y*x^2*exp(x)/(2-y*x^2).
%F A157018 T(n,k) = Product_{m=1..floor(n/2)} binomial(n-2*m,2) = n!/(2^k*(n-2*k)!).
%e A157018 For n = 4 we have 12 lists: 6 1-lists: [{1,2}], [{1,3}], [{1,4}], [{2,3}], [{2,4}], [{3,4}] and 6 2-lists: [{1,2},{3,4}], [{3,4},{1,2}], [{1,3},{2,4}], [{2,4},{1,3}], [{1,4},{2,3}] and [{2,3},{1,4}].
%t A157018 Table[n!/(2^k (n - 2 k)!), {n, 2, 13}, {k, Floor[n/2]}] // Flatten (* _Michael De Vlieger_, Nov 04 2016 *)
%o A157018 (PARI) nmax=100;a=vector(floor(nmax^2/4));idx=0;
%o A157018 for(n=2,nmax,for(k=1,n\2,a[idx++]=n!/(2^k*(n-2*k)!)));
%o A157018 a \\ _Stanislav Sykora_, Nov 03 2016
%Y A157018 Cf. A000262, A000680, A087214, A100861, A126725 (row sums), A129684.
%K A157018 easy,nonn,tabf
%O A157018 2,2
%A A157018 Allan L. Edmonds and _Vladeta Jovovic_, Feb 21 2009