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%I #12 Jul 15 2020 00:19:37
%S 0,1,1,5,13,29,29,93,221,221,733,1757,3805,7901,7901,24285,57053,
%T 122589,122589,384733,384733,384733,2481885,2481885,10870493,10870493,
%U 10870493,10870493,145088221
%N Asymptotic value of odd Catalan numbers mod 2^n.
%C 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.
%H Shu-Chung Liu and Jean C.-C. Yeh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu2/liu6.html">Catalan numbers modulo 2^k</a>, J. Int. Seq. 13 (2010), article 10.5.4.
%F a(n) = remainder(Catalan(2^m-1), 2^n) for any m >= n-1.
%e The odd Catalan numbers mod 2^6=64 are 1,5,45,61,29,29,29, so a(6)=29.
%p A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A038003 := proc(n) A000108(2^n-1) ; end proc:
%p A178854 := proc(n) if n = 0 then 0; else modp(A038003(n-1),2^n) ; end if; end proc:
%p for n from 0 do printf("%d,\n",A178854(n)) ; end do: # _R. J. Mathar_, Jun 28 2010
%t (* 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 *)
%Y Cf. A038003 (odd Catalan numbers).
%K nonn
%O 0,4
%A _David A. Madore_, Jun 18 2010
%E a(12)-a(24) from _Robert G. Wilson v_, Jun 28 2010
%E a(25)-a(28) from _Robert G. Wilson v_, Jul 23 2010