A033536 Cubes of Catalan numbers (A000108).

1, 1, 8, 125, 2744, 74088, 2299968, 78953589, 2924207000, 114933031928, 4738245926336, 203152294091656, 9000469593857728, 410006814589000000, 19129277941464384000, 911218671317138401125, 44202915427981062663000

Offset: 0

Author

N. J. A. Sloane, Dec 10 1999

Keywords

• nonn

Comments

Also the number of maximum independent vertex sets in the 3(n-1)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Dec 31 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Catalan Number.
• Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set.

Crossrefs

Cf. A000108, A001246.

Programs

GAP

List([0..20], n-> (Binomial(2*n, n)/(n+1))^3); # G. C. Greubel, Oct 14 2019

Magma

[Catalan(n)^3: n in [0..20]]; // Vincenzo Librandi, Nov 13 2012

Maple

seq((binomial(2*n, n)/(n+1))^3, n = 0..20); # G. C. Greubel, Oct 14 2019

Mathematica

Table[CatalanNumber@n^3, {n, 0, 20}] (* Vincenzo Librandi, Nov 13 2012 *)
CatalanNumber[Range[0, 20]]^3 (* Eric W. Weisstein, Dec 31 2017 *)

MuPAD

combinat::dyckWords::count(n)^3 $ n = 0..16; // Zerinvary Lajos, Feb 15 2007

PARI

a(n) = (binomial(2*n, n)/(n+1))^3; \\ Altug Alkan, Dec 31 2017

Sage

[catalan_number(i)^3 for i in range(0,17)] # Zerinvary Lajos, May 17 2009

Sage

[catalan_number(n)^3 for n in (0..20)] # G. C. Greubel, Oct 14 2019

Formula

From Ilya Gutkovskiy, Jan 23 2017: (Start)

O.g.f.: (1 - 3F2(-1/2,-1/2,-1/2; 1,1; 64*x))/(8*x).

E.g.f.: 3F3(1/2,1/2,1/2; 2,2,2; 64*x).

a(n) ~ 64^n/(Pi^(3/2)*n^(9/2)). (End)

From Amiram Eldar, Mar 27 2022: (Start)

a(n) = A000108(n)^3.

Sum_{n>=0} a(n)/64^n = 8 - 16*Gamma(3/4)*Gamma(7/4)/(Pi*Gamma(5/4)^2). (End)