A178854 Asymptotic value of odd Catalan numbers mod 2^n.

0, 1, 1, 5, 13, 29, 29, 93, 221, 221, 733, 1757, 3805, 7901, 7901, 24285, 57053, 122589, 122589, 384733, 384733, 384733, 2481885, 2481885, 10870493, 10870493, 10870493, 10870493, 145088221

Offset: 0

Author

David A. Madore, Jun 18 2010

Keywords

• nonn

Comments

For every n, the odd Catalan numbers C(2^m-1) are eventually constant mod 2^n (namely for m >= n-1): then a(n) is the asymptotic value of the remainder.

Examples

			The odd Catalan numbers mod 2^6=64 are 1,5,45,61,29,29,29, so a(6)=29.
		

Links

• Shu-Chung Liu and Jean C.-C. Yeh, Catalan numbers modulo 2^k, J. Int. Seq. 13 (2010), article 10.5.4.

Crossrefs

Cf. A038003 (odd Catalan numbers).

Programs

Maple

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A038003 := proc(n) A000108(2^n-1) ; end proc:
A178854 := proc(n) if n = 0 then 0; else modp(A038003(n-1),2^n) ; end if; end proc:
for n from 0 do printf("%d,\n",A178854(n)) ; end do: # R. J. Mathar, Jun 28 2010

Mathematica

(* first do *) Needs["DiscreteMath`CombinatorialFunctions`"] (* then *) f[n_] := Mod[ CatalanNumber[2^n - 1], 2^n]; Array[f, 25, 0] (* Robert G. Wilson v, Jun 28 2010 *)

Formula

a(n) = remainder(Catalan(2^m-1), 2^n) for any m >= n-1.

Extensions

a(12)-a(24) from Robert G. Wilson v, Jun 28 2010

a(25)-a(28) from Robert G. Wilson v, Jul 23 2010