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A212599 Number of functions on n labeled points to themselves (endofunctions) such that the number of cycles of f that have each even size is even.

%I A212599 #28 Mar 18 2018 17:37:18
%S A212599 1,1,3,18,160,1875,27126,466186,9275064,209654325,5307031000,
%T A212599 148720701426,4570816040352,152874605142727,5527634477245440,
%U A212599 214862754390554250,8934811701563214976,395788795274021394729,18606559519007667893376,925222631836457779380370,48518852386696450625510400
%N A212599 Number of functions on n labeled points to themselves (endofunctions) such that the number of cycles of f that have each even size is even.
%H A212599 Alois P. Heinz, <a href="/A212599/b212599.txt">Table of n, a(n) for n = 0..200</a>
%F A212599 E.g.f.: ((1+T(x))/(1-T(x)))^(1/2) * Product_{i>=1} cosh(T(x)^(2*i)/(2*i)) where T(x) is the e.g.f. for A000169.
%p A212599 with(combinat):
%p A212599 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A212599       add(`if`(irem(j, igcd(i, 2))<>0, 0, (i-1)!^j*
%p A212599       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
%p A212599     end:
%p A212599 a:= n-> add(b(j, j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A212599 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 08 2014
%t A212599 nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];p=Product[Cosh[t^(2i)/(2i)],{i,1,nn}];Range[0,nn]! CoefficientList[Series[((1+t)/(1-t))^(1/2) p,{x,0,nn}],x]
%Y A212599 Cf. A003483, A246951, A116956.
%K A212599 nonn,nice
%O A212599 0,3
%A A212599 _Geoffrey Critzer_, May 22 2012
%E A212599 Maple program fixed by _Vaclav Kotesovec_, Sep 13 2014