A032009 Number of planted planar trees (n+1 nodes) where any 2 subtrees extending from the same node have a different number of nodes.

1, 1, 1, 3, 5, 13, 35, 95, 249, 691, 2007, 5719, 16823, 49371, 146755, 438301, 1319343, 3981699, 12129477, 36987253, 113456615, 348921105, 1077206189, 3332120237, 10347481901, 32183230157, 100372658801, 313633257399, 982232930081, 3080379360481, 9681909324247

Offset

1,4

Comments

Number of compositions of n-1 into distinct parts if there are a(i) kinds of part i. a(6) = 13: 5, 5', 5'', 5''', 5'''', 41, 4'1, 4''1, 14, 14', 14'', 32, 23.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1938

Index entries for sequences related to rooted trees

Formula

Shifts left under "AFK" (ordered, size, unlabeled) transform

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      b(n, i-1, p) +`if`(i>n, 0, a(i)*b(n-i, i-1, p+1))))
    end:
a:= n-> b(n-1$2, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 05 2015

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[ i>n, 0, a[i]*b[n-i, i-1, p+1]]]]; a[n_] := b[n-1, n-1, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

Prog

(PARI)
AFK(v)={apply(p->subst(serlaplace(y^0*p), y, 1), Vec(prod(k=1, #v, 1 + v[k]*x^k*y + O(x*x^#v))))}
seq(n)={my(v=[1]); for(i=1, n, v=AFK(v)); v} \\ Andrew Howroyd, Jun 21 2018

Crossrefs

Cf. A261894.

Sequence in context: A027039 A283844 A324783 * A032027 A360863 A005383

Adjacent sequences: A032006 A032007 A032008 * A032010 A032011 A032012

Keyword

nonn,eigen

Author

Christian G. Bower

Extensions

More terms from Alois P. Heinz, Sep 05 2015

Status

approved