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A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.

%I A204420 #35 Dec 28 2022 01:57:02
%S A204420 1,0,1,0,6,3,0,120,90,15,0,5040,4620,1260,105,0,362880,378000,132300,
%T A204420 18900,945,0,39916800,45571680,18711000,3534300,311850,10395,0,
%U A204420 6227020800,7628100480,3511347840,794593800,94594500,5675670,135135
%N A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
%C A204420 The row polynomials t(n,x):= sum(T(n,k)*x^k, k=0..n) satisfy the recurrence relation t(n,x) = (2n-1)*(x+2n-2)*t(n-1,x), with t(0,x)=1, hence t(n,x)=(2n-1)!!*x(x+2)(x+4)...(x+2n-2).
%H A204420 Alois P. Heinz, <a href="/A204420/b204420.txt">Rows n = 0..85, flattened</a>
%H A204420 Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.
%H A204420 Angelo Lucia and Amanda Young, <a href="https://arxiv.org/abs/2212.11872">A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs</a>, arXiv:2212.11872 [math-ph], 2022.
%F A204420 T(n,k) = (2n-1)!!*2^(n-k)*A132393(n,k).
%F A204420 T(n,k) = (2n-1)T(n-1,k-1) + (2n-1)(2n-2)*T(n-1,k); T(0,0)=1, T(n,0)=0 for n>0,
%F A204420 T(n,n) = (2n-1)!! = A001147(n).
%F A204420 T(n,1) = (2n-1)! = A009445(n-1).
%e A204420 1;
%e A204420 0,        1,
%e A204420 0,        6,        3;
%e A204420 0,      120,       90,       15;
%e A204420 0,     5040,     4620,     1260,     105;
%e A204420 0,   362880,   378000,   132300,   18900,    945;
%e A204420 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;
%p A204420 T_row:= proc(n) local k; seq(doublefactorial(2*n-1)*2^(n-k)* coeff(expand(pochhammer(x, n)), x, k), k=0..n) end: seq(T_row(n), n=0..10);
%t A204420 nn=12;Prepend[Map[Prepend[Select[#,#>0&],0]&,Table[(Range[0, nn]!CoefficientList[ Series[(1-x^2)^(-y/2),{x,0,nn}],{x,y}])[[n]],{n,3,nn,2}]],{1}]//Grid (* _Geoffrey Critzer_, Jul 21 2013 *)
%o A204420 (PARI) T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ _Andrew Howroyd_, Feb 12 2018
%Y A204420 Cf. A049218, A060523, A060524, A132393, A048994.
%Y A204420 Row sums give: A001818. - _Alois P. Heinz_, Jul 21 2013
%K A204420 easy,nonn,tabl
%O A204420 0,5
%A A204420 _José H. Nieto S._, Jan 15 2012