A130644 Number of degree-2n permutations without odd cycles and such that number of cycles of size 2k is odd (or zero) for every k.

1, 1, 6, 225, 8400, 760725, 91725480, 15563633085, 3381661483200, 1015992072520425, 360153767651277600, 160068908768727783825, 84298688029883001074400, 53051020433282263735468125, 38316864396320965168213500000, 32660810942813910822645908353125

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..220

Formula

E.g.f.: Product_{k>0} (1+sinh(x^(2*k)/(2*k))).

Example

a(2)=6 because we have (1234),(1243),(1324),(1342),(1423) and (1432).

Maple

g:=product(1+sinh(x^(2*k)/(2*k)), k=1..50): gser:=series(g, x=0, 44): seq(factorial(2*n)*coeff(gser, x, 2*n), n=0..14); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 and irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015

Mathematica

multinomial[n_, k_] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 && Mod[j, 2] == 1, multinomial[n, Join[{n-i*j}, Array[i&, j]]]*(i-1)!^j/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Crossrefs

Cf. A060307.

Sequence in context: A061610 A054324 A117255 * A223100 A233142 A166502

Adjacent sequences: A130641 A130642 A130643 * A130645 A130646 A130647

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 11 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved