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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

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.

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]

Crossrefs

Cf. A003483, A246951, A116956.

Sequence in context: A371020 A067302 A361048 * A052182 A309985 A328030

Adjacent sequences: A212596 A212597 A212598 * A212600 A212601 A212602

Keyword

nonn,nice

Author

Geoffrey Critzer, May 22 2012

Extensions

Maple program fixed by Vaclav Kotesovec, Sep 13 2014

Status

approved