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
KEYWORD
easy,nonn
AUTHOR
Laurent Orseau, Apr 08 2013
STATUS
approved