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Triangle T(n,k) giving number of degree-2n permutations which decompose into exactly k cycles of even length, k=0..n.
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%I #35 Dec 28 2022 01:57:02

%S 1,0,1,0,6,3,0,120,90,15,0,5040,4620,1260,105,0,362880,378000,132300,

%T 18900,945,0,39916800,45571680,18711000,3534300,311850,10395,0,

%U 6227020800,7628100480,3511347840,794593800,94594500,5675670,135135

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

%C 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 Alois P. Heinz, <a href="/A204420/b204420.txt">Rows n = 0..85, flattened</a>

%H Steven Finch, <a href="https://arxiv.org/abs/2111.14487">Rounds, Color, Parity, Squares</a>, arXiv:2111.14487 [math.CO], 2021.

%H 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 T(n,k) = (2n-1)!!*2^(n-k)*A132393(n,k).

%F 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 T(n,n) = (2n-1)!! = A001147(n).

%F T(n,1) = (2n-1)! = A009445(n-1).

%e 1;

%e 0, 1,

%e 0, 6, 3;

%e 0, 120, 90, 15;

%e 0, 5040, 4620, 1260, 105;

%e 0, 362880, 378000, 132300, 18900, 945;

%e 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;

%p 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 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 (PARI) T(n,k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n,k,1); \\ _Andrew Howroyd_, Feb 12 2018

%Y Cf. A049218, A060523, A060524, A132393, A048994.

%Y Row sums give: A001818. - _Alois P. Heinz_, Jul 21 2013

%K easy,nonn,tabl

%O 0,5

%A _José H. Nieto S._, Jan 15 2012