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

1, 2, 10, 84, 1008, 15840, 308880, 7207200, 196035840, 6094932480, 213322636800, 8303173401600, 355850288640000, 16653793508352000, 845180020548864000, 46236318771202560000, 2712530701243883520000, 169890080762116915200000, 11314679378756986552320000

Offset: 0

Author

Len Smiley, Dec 06 2001

Keywords

• nonn

Comments

From Noam Zeilberger, Mar 19 2019: (Start)

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.)

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)

Examples

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

References

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

Links

• Harry J. Smith, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000108, A001761.

Cf. A370055, A370058.

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

Main diagonal of A256116.

Programs

GAP

List([0..20], n-> 2*Factorial(2*n+1)/Factorial(n+2)); # G. C. Greubel, Mar 19 2019

Magma

[Factorial(n)*Catalan(n+1): n in [0..20]]; // G. C. Greubel, Mar 19 2019

Maple

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
a := n -> (2^(2*n+2)*GAMMA(n+3/2))/(sqrt(Pi)*(n+1)*(n+2)):
seq(simplify(a(n)), n=0..17); # Peter Luschny, Mar 20 2019

Mathematica

Table[2*(2n+1)!/(n+2)!, {n,0,20}] (* G. C. Greubel, Mar 19 2019 *)
Table[n! CatalanNumber[n+1],{n,0,20}] (* Harvey P. Dale, Feb 02 2023 *)

PARI

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

Sage

[factorial(n)*catalan_number(n+1) for n in (0..20)] # G. C. Greubel, Mar 19 2019

Formula

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

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).

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

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

(n+2)*a(n) - 2*n*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Oct 31 2015

a(n) ~ 4^n*exp(-n)*n^(n - 2)*sqrt(2)*(24*n - 61)/6. - Peter Luschny, Mar 20 2019

Sum_{n>=0} 1/a(n) = (25*exp(1/4)*sqrt(Pi)*erf(1/2) + 22)/32, where erf is the error function. - Amiram Eldar, Dec 04 2022

a(n) = 2 * Sum_{k=0..n} (n+2)^(k-1) * |Stirling1(n,k)|. - Seiichi Manyama, Aug 31 2024

E.g.f.: (1/x) * Series_Reversion( x/(1 + x)^2 ). - Seiichi Manyama, Feb 06 2025