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 A224500 Number of ordered full binary trees with labels from a set of at most n labels. 2
 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_{k=1..n} permutations(n, k)*Catalan(k-1); a(n) = Sum_{k=1..n} binomial(n, k)*quadruple_factorial(k-1); a(n) = Sum_{k=1..n} n!(2k-2)!/((n-k)!k!(k-1)!). 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 January 21 00:46 EST 2022. Contains 350473 sequences. (Running on oeis4.)