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.
1, 1, 3, 18, 160, 1875, 27126, 466186, 9275064, 209654325, 5307031000, 148720701426, 4570816040352, 152874605142727, 5527634477245440, 214862754390554250, 8934811701563214976, 395788795274021394729, 18606559519007667893376, 925222631836457779380370, 48518852386696450625510400
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..200
Programs
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Maple
with(combinat): b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(`if`(irem(j, igcd(i, 2))<>0, 0, (i-1)!^j* multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i))) end: a:= n-> add(b(j, j)*n^(n-j)*binomial(n-1, j-1), j=0..n): seq(a(n), n=0..25); # Alois P. Heinz, Sep 08 2014 -
Mathematica
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]
Formula
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.
Extensions
Maple program fixed by Vaclav Kotesovec, Sep 13 2014