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A175137 Irregular triangle T(n,k) read by rows: number of orbits of size 2^k on Dyck n-paths.

%I A175137 #2 Mar 30 2012 17:40:18
%S A175137 1,2,3,1,6,2,1,12,7,4,26,23,11,2,59,71,41,8,138,224,151,30,332,709,
%T A175137 550,114,814,2253,1993,406,16,2028,7189,7211,1564,64,5118,23045,26221,
%U A175137 6010,240,13054,74213,95583,23062,912,33598,239979,349145,88530,3504,87143
%N A175137 Irregular triangle T(n,k) read by rows: number of orbits of size 2^k on Dyck n-paths.
%H A175137 David Callan, <a href="http://www.combinatorics.org/Volume_14/Abstracts/v14i1r28.html">A bijection on Dyck paths and its cycle structure</a>, El. J. Combinat. 14 (2007) # R28
%e A175137 Triangle starts at row n=1
%e A175137 1;
%e A175137 2;
%e A175137 3,1;
%e A175137 6,2,1;
%e A175137 12,7,4;
%e A175137 26,23,11,2;
%e A175137 59,71,41,8;
%e A175137 138,224,151,30;
%p A175137 Fx := proc(k) local ak ; ak := (2*x)^(2^k+1) ; (1-ak-(1-4*x+(ak*x*(2-ak))/(1-x))^(1/2))/(2*x-ak) ; end proc: ff := [] : for k from 0 to 5 do ff := [op(ff), taylor(Fx(k),x=0,18)] ; end do : F := proc(n,k) global ff ; coeftayl(op(k+1,ff),x=0,n) ; end proc: T := proc(n,k) global ff ; if k = 0 then F(n,0) ; else (F(n,k)-F(n,k-1))/2^k ; end if; end proc: for n from 1 to 17 do for k from 0 to 5 do if T(n,k) <> 0 then printf("%d,",T(n,k)) ; fi; end do ; printf("\n") ; end do ;
%Y A175137 Cf. A127384 (row sums).
%K A175137 nonn,tabf
%O A175137 1,2
%A A175137 _R. J. Mathar_, Feb 21 2010