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A224500 Number of ordered full binary trees with labels from a set of at most n labels. 1
1, 4, 21, 184, 2425, 42396, 916909, 23569456, 701312049, 23697421300, 896146948741, 37491632258664, 1719091662617641, 85724109916049164, 4618556912276116125, 267351411229327901536, 16547551265061986364769, 1090506038795558789135076, 76234505063400211010327029 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the maximum number of different operations with n operands for a non-associative non-commutative binary operator.

a(n) is also the second column of A185946.

LINKS

Laurent Orseau, Table for n, a(n) for n=1..350

FORMULA

a(n) = sum(permutations(n, k)*catalan(k-1), k=1..n)

a(n) = sum(binom(n, k)*quadruple_factorial(k-1), k=1..n)

a(n) = sum(n!(2k-2)!/((n-k)!k!(k-1)!), k=1..n)

a(1)=1, a(2)=4, a(n) = (4n-5)*a(n-1)-(4n-4)*a(n-2)+1 for n>2. - Giovanni Resta, Apr 08 2013

E.g.f: exp(x)*(1-sqrt(1-4*x))/2  - Mark van Hoeij, Apr 10 2013

G.f.: hypergeom([1,1/2],[],4*x/(1-x))*x/(1-x)^2  - Mark van Hoeij, Apr 10 2013

a(n) ~ 2^(2*n-3/2)*n^(n-1)*exp(1/4-n). - Vaclav Kotesovec, Aug 16 2013

EXAMPLE

For n=3, the a(3)=21 solutions are:

    a b c

    ab ba ac ca bc cb

    (ab)c a(bc)

    (ac)b a(cb)

    (ba)c b(ac)

    (bc)a b(ca)

    (ca)b c(ab)

    (cb)a c(ba)

MATHEMATICA

a[n_] := Sum[Binomial[n, k]*(2*k-2)! / (k-1)!, {k, n}]; Array[a, 20] (* Giovanni Resta, Apr 08 2013 *)

PROG

(Racket)

#lang racket

(require math/number-theory)

(define (a n)

  (for/sum ([k (in-range 1 (+ n 1))])

    (* (binomial n k)

       (/ (factorial (* 2 (- k 1)))

          (factorial (- k 1))))))

(PARI) x='x+O('x^66); Vec(serlaplace(exp(x)*(1-sqrt(1-4*x))/2)) /* Joerg Arndt, Apr 10 2013 */

CROSSREFS

Cf. A185946, A220452, A001813, A000108

Sequence in context: A231220 A231434 A221370 * A158108 A158258 A065527

Adjacent sequences:  A224497 A224498 A224499 * A224501 A224502 A224503

KEYWORD

easy,nonn

AUTHOR

Laurent Orseau, Apr 08 2013

STATUS

approved

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Last modified June 25 11:31 EDT 2017. Contains 288709 sequences.