cp's OEIS Frontend

A224500 Number of ordered full binary trees with labels from a set of at most n labels.

%I A224500 #27 Jan 05 2021 00:04:02
%S A224500 1,4,21,184,2425,42396,916909,23569456,701312049,23697421300,
%T A224500 896146948741,37491632258664,1719091662617641,85724109916049164,
%U A224500 4618556912276116125,267351411229327901536,16547551265061986364769,1090506038795558789135076,76234505063400211010327029
%N A224500 Number of ordered full binary trees with labels from a set of at most n labels.
%C A224500 a(n) is also the maximum number of different operations with n operands for a non-associative non-commutative binary operator.
%C A224500 a(n) is also the second column of A185946.
%H A224500 Laurent Orseau, <a href="/A224500/b224500.txt">Table for n, a(n) for n = 1..350</a>
%F A224500 a(n) = Sum_{k=1..n} permutations(n, k)*Catalan(k-1);
%F A224500 a(n) = Sum_{k=1..n} binomial(n, k)*quadruple_factorial(k-1);
%F A224500 a(n) = Sum_{k=1..n} n!(2k-2)!/((n-k)!k!(k-1)!).
%F A224500 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
%F A224500 E.g.f.: exp(x)*(1-sqrt(1-4*x))/2. - _Mark van Hoeij_, Apr 10 2013
%F A224500 G.f.: hypergeom([1,1/2],[],4*x/(1-x))*x/(1-x)^2. - _Mark van Hoeij_, Apr 10 2013
%F A224500 a(n) ~ 2^(2*n-3/2)*n^(n-1)*exp(1/4-n). - _Vaclav Kotesovec_, Aug 16 2013
%e A224500 For n=3, the a(3)=21 solutions are:
%e A224500     a b c
%e A224500     ab ba ac ca bc cb
%e A224500     (ab)c a(bc)
%e A224500     (ac)b a(cb)
%e A224500     (ba)c b(ac)
%e A224500     (bc)a b(ca)
%e A224500     (ca)b c(ab)
%e A224500     (cb)a c(ba)
%t A224500 a[n_] := Sum[Binomial[n,k]*(2*k-2)! / (k-1)!, {k,n}]; Array[a,20] (* _Giovanni Resta_, Apr 08 2013 *)
%o A224500 (Racket)
%o A224500 #lang racket
%o A224500 (require math/number-theory)
%o A224500 (define (a n)
%o A224500   (for/sum ([k (in-range 1 (+ n 1))])
%o A224500     (* (binomial n k)
%o A224500        (/ (factorial (* 2 (- k 1)))
%o A224500           (factorial (- k 1))))))
%o A224500 (PARI) x='x+O('x^66); Vec(serlaplace(exp(x)*(1-sqrt(1-4*x))/2)) /* _Joerg Arndt_, Apr 10 2013 */
%Y A224500 Cf. A185946, A220452, A001813, A000108.
%K A224500 easy,nonn
%O A224500 1,2
%A A224500 _Laurent Orseau_, Apr 08 2013