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A033536 Cubes of Catalan numbers (A000108).

%I A033536 #37 Feb 16 2025 08:32:36
%S A033536 1,1,8,125,2744,74088,2299968,78953589,2924207000,114933031928,
%T A033536 4738245926336,203152294091656,9000469593857728,410006814589000000,
%U A033536 19129277941464384000,911218671317138401125,44202915427981062663000
%N A033536 Cubes of Catalan numbers (A000108).
%C A033536 Also the number of maximum independent vertex sets in the 3(n-1)-triangular honeycomb acute knight graph. - _Eric W. Weisstein_, Dec 31 2017
%H A033536 Vincenzo Librandi, <a href="/A033536/b033536.txt">Table of n, a(n) for n = 0..500</a>
%H A033536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalanNumber.html">Catalan Number</a>.
%H A033536 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumIndependentVertexSet.html">Maximum Independent Vertex Set</a>.
%F A033536 From _Ilya Gutkovskiy_, Jan 23 2017: (Start)
%F A033536 O.g.f.: (1 - 3F2(-1/2,-1/2,-1/2; 1,1; 64*x))/(8*x).
%F A033536 E.g.f.: 3F3(1/2,1/2,1/2; 2,2,2; 64*x).
%F A033536 a(n) ~ 64^n/(Pi^(3/2)*n^(9/2)). (End)
%F A033536 From _Amiram Eldar_, Mar 27 2022: (Start)
%F A033536 a(n) = A000108(n)^3.
%F A033536 Sum_{n>=0} a(n)/64^n = 8 - 16*Gamma(3/4)*Gamma(7/4)/(Pi*Gamma(5/4)^2). (End)
%p A033536 seq((binomial(2*n, n)/(n+1))^3, n = 0..20); # _G. C. Greubel_, Oct 14 2019
%t A033536 Table[CatalanNumber@n^3, {n, 0, 20}] (* _Vincenzo Librandi_, Nov 13 2012 *)
%t A033536 CatalanNumber[Range[0, 20]]^3 (* _Eric W. Weisstein_, Dec 31 2017 *)
%o A033536 (MuPAD) combinat::dyckWords::count(n)^3 $ n = 0..16; // _Zerinvary Lajos_, Feb 15 2007
%o A033536 (Sage) [catalan_number(i)^3 for i in range(0,17)] # _Zerinvary Lajos_, May 17 2009
%o A033536 (Magma) [Catalan(n)^3: n in [0..20]]; // _Vincenzo Librandi_, Nov 13 2012
%o A033536 (PARI) a(n) = (binomial(2*n, n)/(n+1))^3; \\ _Altug Alkan_, Dec 31 2017
%o A033536 (Sage) [catalan_number(n)^3 for n in (0..20)] # _G. C. Greubel_, Oct 14 2019
%o A033536 (GAP) List([0..20], n-> (Binomial(2*n, n)/(n+1))^3); # _G. C. Greubel_, Oct 14 2019
%Y A033536 Cf. A000108, A001246.
%K A033536 nonn
%O A033536 0,3
%A A033536 _N. J. A. Sloane_, Dec 10 1999