OFFSET
0,3
COMMENTS
From Piet Hut, Nov 07 2003: "Number of ways to place n stars in stable hierarchical multiple star systems (where each stable multiple is a binary tree: around its center of mass two multiple star systems revolve, each of which can be a singleton or a nontrivial multiple star system).
"For example, a(1) = 1 : *; a(2) = 2 : (**), * *; a(3) = 3 : ((**)*), (**) *, * * *; a(4) = 6 : (((**)*)*), ((**)(**)), ((**)*) *, (**) (**), (**) * *, * * * * ."
REFERENCES
L. F. Meyers, Corrections and additions to Tree Representations in Linguistics. Report 3, 1966, p. 138. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
L. F. Meyers and W. S.-Y. Wang, Tree Representations in Linguistics. Report 3, 1963, pp. 107-108. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
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 = 0..2544 (first 201 terms from T. D. Noe)
Piet Hut, Home Page.
FORMULA
Euler transform of A001190. - Michael Somos, Nov 10 2003
G.f.: exp( Sum_{i>=1} G001190(x^i)/i ), where G001190 = g.f. for A001190.
a(n) ~ c * d^n / n^(3/2), where d = A086317 = 2.4832535361726368585622885181... and c = 0.9874010699028009804... . - Vaclav Kotesovec, Apr 19 2016
MAPLE
b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
(t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d),
d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Sep 11 2017
MATHEMATICA
terms = 35; (* G = G001190 *) G[_] = 0; Do[G[x_] = x + (1/2)*(G[x]^2 + G[x^2]) + O[x]^terms // Normal, terms]; A[x_] = Exp[Sum[G[x^i]/i, {i, 1, terms}]] + O[x]^terms; CoefficientList[A[x], x](* Jean-François Alcover, Nov 18 2011, updated Jan 12 2018 *)
(* b = A001190 *) b[n_] := b[n] = If[OddQ[n], Sum[b[k] b[n-k], {k, 1, (n-1)/2}], Sum[b[k] b[n-k], {k, 1, n/2 - 1}] + (1/2) b[n/2] (1+b[n/2])]; b[0] = 0; b[1] = 1;
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[d p[d], {d, Divisors[j]}] b[n-j], {j, 1, n}]/n]; b];
a[n_] := etr[b][n]; Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Mar 14 2016 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved