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A065866 a(n) = n! * Catalan(n+1). 4

%I

%S 1,2,10,84,1008,15840,308880,7207200,196035840,6094932480,

%T 213322636800,8303173401600,355850288640000,16653793508352000,

%U 845180020548864000,46236318771202560000,2712530701243883520000,169890080762116915200000,11314679378756986552320000

%N a(n) = n! * Catalan(n+1).

%C From _Noam Zeilberger_, Mar 19 2019: (Start)

%C a(n) is the number of flags in the associahedron of dimension n. For example, there are a(2) = 10 flags in the associahedron of dimension 2, a pentagon. (In this case a flag corresponds to a triple v:e:f of a mutually incident vertex v, edge e, and face f, with f necessarily the unique face of the pentagon.)

%C Equivalently, a(n) is the number of maximal sequences of consistent parenthesizations of a string of n + 2 letters, starting with n + 1 pairs of parentheses, then removing one pair, and so on, ending with the trivial (outermost) parenthesization. For example, (a(b(cd))):(ab(cd)):(abcd) and (a(b(cd))):(a(bcd)):(abcd) are two of the a(2) = 10 maximal sequences of consistent parenthesizations of the letters abcd. (End)

%D R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, pp. 200-204.

%H Harry J. Smith, <a href="/A065866/b065866.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = 2 * (2n+1)!/(n+2)!.

%F E.g.f.: (1-2*x-sqrt(1-4*x))/(2*x^2) = (O.g.f. for A000108)^2 = B_2(x)^2 (cf. GKP reference).

%F 0 = a(n)*(-7200*a(n+2) + 2700*a(n+3) + 246*a(n+4) - 33*a(n+5)) + a(n+1)*(+36*a(n+2) + 372*a(n+3) + 36*a(n+4) - a(n+5)) + a(n+2)*(-18*a(n+2) + 9*a(n+3) + a(n+4)) for n >= 0. - _Michael Somos_, Apr 14 2015

%F The e.g.f. A(x) satisfies 0 = -2 + A(x) * (6*x - 2) + A'(x) * (4*x^2 - x). - _Michael Somos_, Apr 14 2015

%F Conjecture: (n+2)*a(n) - 2*n*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Oct 31 2015

%F a(n) ~ 4^n*exp(-n)*n^(n - 2)*sqrt(2)*(24*n - 61)/6. - _Peter Luschny_, Mar 20 2019

%e G.f. = 1 + 2*x + 10*x^2 + 84*x^3 + 1008*x^4 + 15840*x^5 + 308880*x^6 + ...

%p with(combstruct): ZL:=[T, {T=Union(Z, Prod(Epsilon, Z, T), Prod(T, Z, Epsilon), Prod(T, T, Z))}, labeled]: seq(count(ZL, size=i+1)/(i+1), i=0..18); # _Zerinvary Lajos_, Dec 16 2007

%p a := n -> (2^(2*n+2)*GAMMA(n+3/2))/(sqrt(Pi)*(n+1)*(n+2)):

%p seq(simplify(a(n)), n=0..17); # _Peter Luschny_, Mar 20 2019

%t Table[2*(2n+1)!/(n+2)!, {n,0,20}] (* _G. C. Greubel_, Mar 19 2019 *)

%o (PARI) { for (n = 0, 100, a = 2 * (2*n + 1)!/(n + 2)!; write("b065866.txt", n, " ", a) ) } \\ _Harry J. Smith_, Nov 02 2009

%o (MAGMA) [Factorial(n)*Catalan(n+1): n in [0..20]]; // _G. C. Greubel_, Mar 19 2019

%o (Sage) [factorial(n)*catalan_number(n+1) for n in (0..20)] # _G. C. Greubel_, Mar 19 2019

%o (GAP) List([0..20], n-> 2*Factorial(2*n+1)/Factorial(n+2)) # _G. C. Greubel_, Mar 19 2019

%Y Cf. A000108.

%Y Equals 2 * A102693(n+1), n > 0.

%Y Main diagonal of A256116.

%K nonn

%O 0,2

%A _Len Smiley_, Dec 06 2001

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Last modified May 25 11:17 EDT 2019. Contains 323539 sequences. (Running on oeis4.)