OFFSET
1,2
COMMENTS
Also labeled involution rooted trees.
REFERENCES
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).
LINKS
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
FORMULA
E.g.f. A(x) satisfies A(x) = x*exp(A(x)+A(x)^2/2).
a(n) = i^(n-1)*n^((n-1)/2)*He_{n-1}(-sqrt(-n)), i=sqrt(-1), He_k unitary Hermite polynomial (cf. A066325).
a(n) = Sum_{k = ceiling((n-1)/2)...n-1} (n-1)!/((n-k-1)!*(2*k-n+1)!)*n^k*2^(-n+k+1). - Vladimir Kruchinin, Aug 07 2012
a(n) ~ 2^(n+1/2) * n^(n-1) * exp((sqrt(5)-3)*n/4) / (sqrt(5+sqrt(5)) * (sqrt(5)-1)^n). - Vaclav Kotesovec, Jan 08 2014
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[x/E^(x*(2+x)/2), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
PROG
(Maxima) a(n):=sum(((n-1)!/((n-k-1)!*(2*k-n+1)!)*n^k*2^(-n+k+1)), k, ceiling((n-1)/2), n-1); /* Vladimir Kruchinin, Aug 07 2012 */
(PARI) x='x+O('x^66);
Vec(serlaplace(serreverse(x/exp(x^2/2+x)))) /* Joerg Arndt, Jan 25 2013 */
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
Christian G. Bower, Jan 13 2004
STATUS
approved