A032301 Shifts left under "EFJ" (unordered, size, labeled) transform.

1, 1, 1, 4, 8, 38, 206, 1200, 6824, 50912, 446752, 3828592, 38953680, 411358960, 4740541440, 57933236928, 759535226432, 10488778719488, 156933187370432, 2425018017191040, 40031753222399360, 689218695990369536, 12461424512466701312, 234386152841716303616

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

Sequence in context: A231398 A231465 A208820 * A032213 A225824 A032317

Adjacent sequences: A032298 A032299 A032300 * A032302 A032303 A032304

Keyword

nonn,eigen

Author

Christian G. Bower

Status

approved