A216187 Number of labeled rooted trees on n nodes such that each internal node has an odd number of children.

0, 1, 2, 6, 28, 200, 1926, 22512, 306104, 4770432, 84234250, 1663735040, 36320155092, 867963393024, 22535294920334, 631718010255360, 19016907901995376, 611869203759792128, 20954324710009221138, 761015341362413371392, 29214930870257449355660

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Formula

E.g.f. satisfies: F(x) = x*(sinh(F(x))+1).

a(n) ~ sqrt(s/(s-r)) * n^(n-1) / (exp(n) * r^n), where r = 0.482309923717218507261475229723265094762759829863... and s = 1.358310572965774067065006624540704170183889018218... are real roots of the system of equations s = r*(1 + sinh(s)), r*cosh(s) = 1. - Vaclav Kotesovec, Jun 07 2021

Example

a(5) = 200: There are three unlabeled rooted trees of 5 nodes with all internal nodes having an odd number of children. They can be labeled respectively in 20 + 120 + 60 = 200 ways.
  ..o............o............o....
  ..|............|.........../|\...
  ..o............o..........o.o.o..
  ./|\...........|..........|......
  o.o.o..........o..........o......
  ...............|.................
  ...............o.................
  ...............|.................
  ...............o.................

Maple

a:= n-> n!*coeff(series(RootOf(F=x*(sinh(F)+1), F), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 12 2013

Mathematica

nn=12; f[x_]:=Sum[a[n]x^n/n!, {n, 0, nn}]; s=SolveAlways[0==Series[f[x]-x (Sinh[f[x]]+1), {x, 0, nn}], x]; Table[a[n], {n, 0, nn}]/.s

Crossrefs

Cf. A036778.

Sequence in context: A058128 A229112 A201959 * A178446 A324126 A272662

Adjacent sequences: A216184 A216185 A216186 * A216188 A216189 A216190

Keyword

nonn

Author

Geoffrey Critzer, Mar 11 2013

Status

approved