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
Keywords
Examples
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.................
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..150
Crossrefs
Cf. A036778.
Programs
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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
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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
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