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A204420 Triangle T(n,k) giving number of degree-2n permutations which decompose into k cycles of even length, k=0..n. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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).

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

LINKS

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

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).

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: A153459 A102525 A119923 * 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

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Last modified June 1 07:40 EDT 2020. Contains 334759 sequences. (Running on oeis4.)