OFFSET
1,4
COMMENTS
a(n) is the number of increasing rooted trees where any 2 subtrees extending from the same node have a different number of nodes (the unlabeled trees counted by A032305). An increasing tree is labeled so that every path from the root to an external node is increasing. - Geoffrey Critzer, Jul 29 2013
(a(n)/n!)^(1/n) tends to 0.82143368... - Vaclav Kotesovec, Jul 21 2019
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..200
Vaclav Kotesovec, Plot of a(n+1)/a(n)/n for n = 1..3300
FORMULA
E.g.f.: A(x) satisfies: A'(x) = Product_{n>=1} 1 + a(n) x^n/n!. - Geoffrey Critzer, Jul 29 2013
MAPLE
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, i$j, n-i*j)*binomial(b((i-1)$2), j)
*b(n-i*j, i-1), j=0..min(1, n/i))))
end:
a:= n-> b((n-1)$2):
seq(a(n), n=1..30); # Alois P. Heinz, Jul 31 2013
MATHEMATICA
nn=15; f[x_]:=Sum[a[n]x^n/n!, {n, 0, nn}]; sol=SolveAlways[0==Series[f[x] -Integrate[Product[1+a[i]x^i/i!, {i, 1, nn}], x], {x, 0, nn}], x]; Table[a[n], {n, 0, nn}]/.sol (* Geoffrey Critzer, Jul 29 2013 *)
PROG
(PARI) EFJ(v)={Vec(serlaplace(prod(k=1, #v, 1 + v[k]*x^k/k! + O(x*x^#v)))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], EFJ(v))); v} \\ Andrew Howroyd, Sep 11 2018
CROSSREFS
KEYWORD
nonn,eigen
AUTHOR
STATUS
approved