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

1, 0, 1, 0, 6, 3, 0, 120, 90, 15, 0, 5040, 4620, 1260, 105, 0, 362880, 378000, 132300, 18900, 945, 0, 39916800, 45571680, 18711000, 3534300, 311850, 10395, 0, 6227020800, 7628100480, 3511347840, 794593800, 94594500, 5675670, 135135

Offset

0,5

Comments

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).

Links

Alois P. Heinz, Rows n = 0..85, flattened

Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Angelo Lucia and Amanda Young, A Nonvanishing Spectral Gap for AKLT Models on Generalized Decorated Graphs, arXiv:2212.11872 [math-ph], 2022.

Formula

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

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,

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

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

Example

1;
0,        1,
0,        6,        3;
0,      120,       90,       15;
0,     5040,     4620,     1260,     105;
0,   362880,   378000,   132300,   18900,    945;
0, 39916800, 45571680, 18711000, 3534300, 311850, 10395;

Maple

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);

Mathematica

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 *)

Prog

(PARI) T(n, k) = (2*n)!/(2^n*n!)*(-2)^(n-k)*stirling(n, k, 1); \\ Andrew Howroyd, Feb 12 2018

Crossrefs

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

Row sums give: A001818. - Alois P. Heinz, Jul 21 2013

Sequence in context: A102525 A119923 A359194 * A331570 A102410 A105123

Adjacent sequences: A204417 A204418 A204419 * A204421 A204422 A204423

Keyword

easy,nonn,tabl

Author

José H. Nieto S., Jan 15 2012

Status

approved