A006964 Number of directed rooted trees with n nodes.
1, 3, 15, 82, 495, 3144, 20875, 142773, 1000131, 7136812, 51702231, 379234623, 2810874950, 21020047557, 158398829121, 1201617201230, 9169060501023, 70329406653879, 541949364313821, 4193569906262874, 32571403998781956, 253842927519362734, 1984442128649393178
Offset: 1
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..500
- Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 439
- P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Mathh
- . Québec, Vol. 16, No. 1 (1992) pp. 53-80.
- P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
- R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
- Index entries for sequences related to rooted trees
- Index entries for sequences related to trees
Programs
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Maple
with(numtheory): a:= proc(n) option remember; `if`(n<2, n, (add(add(d*a(d), d=divisors(j)) *a(n-j)*3, j=1..n-1))/(n-1)) end: seq(a(n), n=1..30); # Alois P. Heinz, Sep 06 2008
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Mathematica
a[n_] := a[n] = If[n<2, n, (Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j]*3, {j, 1, n-1}])/(n-1)]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)
Formula
a(n+1) has g.f.: Product_{n>=1} (1 - x^3*a(n))^-1.
a(n) ~ c * d^n / n^(3/2), where d = 8.356026879295995368276069578708912..., c = 0.13645899548680457355557420025756... . - Vaclav Kotesovec, Aug 20 2014
G.f. A(x) satisfies: A(x) = x*exp(3*Sum_{k>=1} A(x^k)/k). - Ilya Gutkovskiy, Mar 19 2018
Extensions
Extended by Christian G. Bower, Apr 15 1998
Comments