A033536 Cubes of Catalan numbers (A000108).

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

Offset

0,3

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.

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)

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 *)

Prog

(MuPAD) combinat::dyckWords::count(n)^3 $ n = 0..16; // Zerinvary Lajos, Feb 15 2007
(Sage) [catalan_number(i)^3 for i in range(0, 17)] # Zerinvary Lajos, May 17 2009
(Magma) [Catalan(n)^3: n in [0..20]]; // Vincenzo Librandi, Nov 13 2012
(PARI) a(n) = (binomial(2*n, n)/(n+1))^3; \\ Altug Alkan, Dec 31 2017
(Sage) [catalan_number(n)^3 for n in (0..20)] # G. C. Greubel, Oct 14 2019
(GAP) List([0..20], n-> (Binomial(2*n, n)/(n+1))^3); # G. C. Greubel, Oct 14 2019

Crossrefs

Cf. A000108, A001246.

Sequence in context: A264143 A110272 A215040 * A355762 A215793 A273279

Adjacent sequences: A033533 A033534 A033535 * A033537 A033538 A033539

Keyword

nonn

Author

N. J. A. Sloane, Dec 10 1999

Status

approved