A093637 G.f.: A(x) = Product_{n>=0} 1/(1 - a(n)*x^(n+1)) = Sum_{n>=0} a(n)*x^n.

1, 1, 2, 4, 9, 20, 49, 117, 297, 746, 1947, 5021, 13378, 35237, 95123, 254825, 694987, 1882707, 5184391, 14177587, 39289183, 108337723, 301997384, 837774846, 2347293253, 6546903307, 18417850843, 51617715836, 145722478875, 409964137081, 1161300892672

Offset: 0

Author

Paul D. Hanna, Apr 07 2004

Keywords

• nonn

Comments

From David Callan, Nov 02 2006: (Start)

a(n) = number of (unlabeled, rooted) ordered trees on n edges such that, for each vertex of outdegree >= 1, the sizes of its subtrees are weakly increasing left to right. This notion is close to that of unlabeled, unordered rooted tree (A000081) but, for example,

./\...../\.

|./\.../\.|

|.........|

count as two different trees here whereas A000081 treats them as the same.

(End)

We can also think of a(n) in terms of integer partitions, recursively: Let a(0)=1. For each partition n=p1+p2+p3+...+pr, consider the number q=a(p1-1)*a(p2-1)*...*a(pr-1). Then, summing these q over all the partitions of n gives a(n). - Daniele P. Morelli, May 22 2010

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 49*x^6 +...
where
A(x) = 1/((1-x)*(1-x^2)*(1-2*x^3)*(1-4*x^4)*(1-9*x^5)*(1-20*x^6)*(1-49*x^7)...).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000081, A093635, A093638.

Programs

Maple

b:= proc(n, i) option remember; `if`(i>n, 0,
       a(i-1)*`if`(i=n, 1, b(n-i, i)))+`if`(i>1, b(n, i-1), 0)
    end:
a:= n-> `if`(n=0, 1, b(n, n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 20 2012

Mathematica

b[n_, i_] := b[n, i] = If[i>n, 0, a[i-1]*If[i == n, 1, b[n-i, i]]] + If[i>1, b[n, i-1], 0]; a[n_] := If[n == 0, 1, b[n, n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 15 2015, after Alois P. Heinz *)

PARI

{a(n) = polcoeff(prod(i=0,n-1,1/(1-a(i)*x^(i+1)))+x*O(x^n),n)}
for(n=0,25,print1(a(n),", "))

PARI

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,1/m*sum(k=1,n, polcoeff(A+O(x^k), k-1)^m*x^(m*k)) +x*O(x^n))));polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = exp( Sum_{n>=1} Sum_{k>=1} a(k)^n * (x^k)^n /n ). - Paul D. Hanna, Oct 26 2011