A167893 a(n) = Sum_{k=1..n} Catalan(k)^3.

1, 9, 134, 2878, 76966, 2376934, 81330523, 3005537523, 117938569451, 4856184495787, 208008478587443, 9208478072445171, 419215292661445171, 19548493234125829171, 930767164551264230296, 45133682592532326893296, 2224173698690413601132296, 111192059034974606204132296

Offset: 1

Author

Alexander Adamchuk, Nov 15 2009

Keywords

• nonn

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

Crossrefs

Cf. A000108, A014138, A167892, A167893, A001246, A033536, A014137, A094639.

Programs

Magma

[&+[Catalan(i)^3: i in [1..n]]: n in [1..20]]; // Vincenzo Librandi, 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 *)

PARI

a(n)=sum(k=1,n,(binomial(k+k,k)/(k+1))^3) /* Charles R Greathouse IV, Jun 14 2011 */

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

Extensions

More terms from J. Mulder, (jasper.mulder(AT)planet.nl), Jan 25 2010

More terms from Sean A. Irvine, Jun 13 2011