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A130648 Number of degree-n permutations without even cycles and such that number of cycles of size 2k-1 is odd (or zero) for every k. 2

%I

%S 1,1,0,3,8,25,184,721,9904,66753,691088,5973121,84925048,940427137,

%T 12801319816,186556383105,3174772979936,48489077948161,

%U 842173637012896,15359492773456129,316965131969908072,6368424993521096961,135098381153771956952,2980219360336428021505

%N Number of degree-n permutations without even cycles and such that number of cycles of size 2k-1 is odd (or zero) for every k.

%H Alois P. Heinz, <a href="/A130648/b130648.txt">Table of n, a(n) for n = 0..450</a>

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

%e a(3)=3 because we have (1)(2)(3), (123) and (132).

%p g:=product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30): gser:=series(g,x=0,27): seq(factorial(n)*coeff(gser,x,n),n=0..24); # _Emeric Deutsch_, Aug 24 2007

%p # second Maple program:

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

%p `if`(j=0 or irem(i, 2)=1 and irem(j, 2)=1, multinomial(n,

%p n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))

%p end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 09 2015

%t 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] == 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-Fran├žois Alcover_, Feb 08 2017, after _Alois P. Heinz_ *)

%Y Cf. A060307.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Aug 11 2007

%E More terms from _Emeric Deutsch_, Aug 24 2007

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Last modified November 22 03:34 EST 2019. Contains 329385 sequences. (Running on oeis4.)