OFFSET
1,2
COMMENTS
Catalan(k) = A000108(k) = (2k)!/(k!*(k+1)!) = C(2*k,k)/(k+1).
For prime p=7, p^2 divides a(p^2), and p divides all a(n) for n from (p^2-1)/2 to p^2-2.
For prime p=19 or 97, p divides all a(n) for n from (p-1)/2 to p-2.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Catalan Number
FORMULA
a(n) = Sum_{k=1..n} A033536(k).
Recurrence: (n+1)^3*a(n) = (5*n - 1)*(13*n^2 - 16*n + 7)*a(n-1) - 8*(2*n - 1)^3*a(n-2). - Vaclav Kotesovec, Jul 01 2016
a(n) ~ 2^(6*n+6) / (63*Pi^(3/2)*n^(9/2)). - Vaclav Kotesovec, Jul 01 2016
MATHEMATICA
Array[n \[Function] Sum[CatalanNumber[k]^3, {k, 1, n}], 15] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)
Accumulate[CatalanNumber[Range[1, 20]]^3] (* Vincenzo Librandi, Jul 01 2016 *)
PROG
(PARI) a(n)=sum(k=1, n, (binomial(k+k, k)/(k+1))^3) /* Charles R Greathouse IV, Jun 14 2011 */
(Magma) [&+[Catalan(i)^3: i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, Jul 01 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Nov 15 2009
EXTENSIONS
More terms from J. Mulder, (jasper.mulder(AT)planet.nl), Jan 25 2010
More terms from Sean A. Irvine, Jun 13 2011
STATUS
approved