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A093637
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G.f.: A(x) = Product_{n>=0} 1/(1-a(n)*x^(n+1)) = Sum_{n>=0} a(n)*x^n.
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Comment from David Callan, Nov 02 2006: 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.
Contribution from Daniele P. Morelli (netherself(AT)gmail.com), May 22 2010: (Start)
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). (End)
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FORMULA
| G.f. satisfies: A(x) = exp( Sum_{n>=1} Sum_{k>=1} a(k)^n * (x^k)^n /n ). [From Paul D. Hanna, Oct 26 2011]
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EXAMPLE
| 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)...).
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PROG
| (PARI) a(n) = polcoeff(prod(i=0, n-1, 1/(1-a(i)*x^(i+1)))+x*O(x^n), 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)}
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CROSSREFS
| Cf. A000081, A093635, A093638.
Sequence in context: A145550 A000081 A124497 * A068051 A032289 A006648
Adjacent sequences: A093634 A093635 A093636 * A093638 A093639 A093640
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Apr 07 2004
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