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A216187
Number of labeled rooted trees on n nodes such that each internal node has an odd number of children.
1
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
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.
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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
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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Mar 11 2013
STATUS
approved