A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

1, 1, 2, 6, 21, 105, 675, 4725, 35805, 322245, 3236625, 35602875, 425872755, 5536345815, 77347084815, 1160206272225, 18403556596425, 312860462139225, 5643104418376425, 107218983949152075, 2136610763952639975, 44868826043005439475, 986129980012277775675

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

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

Example

a(4)=21 because only the following three degree-4 permutations do not qualify: (12)(34), (13)(24) and (14)(23).

Maple

g:=sqrt((1+x)/(1-x))*(product(1+sinh(x^(2*k)/(2*k)), k=1..30)): gser:=series(g, x=0, 25): seq(factorial(n)*coeff(gser, x, n), n=0..20); # 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)=1 or 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(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 || 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[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Crossrefs

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

Sequence in context: A008987 A079129 A305923 * A363362 A156808 A245882

Adjacent sequences: A130272 A130273 A130274 * A130276 A130277 A130278

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 06 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved