A003214 Number of binary forests with n nodes. (Formerly M0775)

1, 1, 2, 3, 6, 10, 20, 37, 76, 152, 320, 672, 1454, 3154, 6959, 15439, 34608, 77988, 176985, 403510, 924683, 2127335, 4913452, 11385955, 26468231, 61700232, 144206269, 337837221, 793213550, 1866181155, 4398867672, 10387045476, 24567374217, 58196129468, 138056734916

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

Cf. A001190.

Sequence in context: A007148 A093371 A339153 * A331693 A123423 A005195

Adjacent sequences: A003211 A003212 A003213 * A003215 A003216 A003217

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved