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%I M0775 #36 Sep 02 2023 18:52:34
%S 1,1,2,3,6,10,20,37,76,152,320,672,1454,3154,6959,15439,34608,77988,
%T 176985,403510,924683,2127335,4913452,11385955,26468231,61700232,
%U 144206269,337837221,793213550,1866181155,4398867672,10387045476,24567374217,58196129468,138056734916
%N Number of binary forests with n nodes.
%C 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).
%C "For example, a(1) = 1 : *; a(2) = 2 : (**), * *; a(3) = 3 : ((**)*), (**) *, * * *; a(4) = 6 : (((**)*)*), ((**)(**)), ((**)*) *, (**) (**), (**) * *, * * * * ."
%D 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.
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A003214/b003214.txt">Table of n, a(n) for n = 0..2544</a> (first 201 terms from T. D. Noe)
%H Piet Hut, <a href="http://www.sns.ias.edu/~piet/">Home Page</a>.
%F Euler transform of A001190. - _Michael Somos_, Nov 10 2003
%F G.f.: exp( Sum_{i>=1} G001190(x^i)/i ), where G001190 = g.f. for A001190.
%F a(n) ~ c * d^n / n^(3/2), where d = A086317 = 2.4832535361726368585622885181... and c = 0.9874010699028009804... . - _Vaclav Kotesovec_, Apr 19 2016
%p b:= proc(n) option remember; `if`(n<2, n, `if`(n::odd, 0,
%p (t-> t*(1-t)/2)(b(n/2)))+add(b(i)*b(n-i), i=1..n/2))
%p end:
%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d),
%p d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Sep 11 2017
%t 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 *)
%t (* 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;
%t 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];
%t a[n_] := etr[b][n]; Table[a[n], {n, 0, 34}] (* _Jean-François Alcover_, Mar 14 2016 *)
%Y Cf. A001190.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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