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

1, 2, 3, 1, 6, 2, 1, 12, 7, 4, 26, 23, 11, 2, 59, 71, 41, 8, 138, 224, 151, 30, 332, 709, 550, 114, 814, 2253, 1993, 406, 16, 2028, 7189, 7211, 1564, 64, 5118, 23045, 26221, 6010, 240, 13054, 74213, 95583, 23062, 912, 33598, 239979, 349145, 88530, 3504, 87143

Offset: 1

Author

R. J. Mathar, Feb 21 2010

Keywords

• nonn
• tabf

Examples

			Triangle starts at row n=1
1;
2;
3,1;
6,2,1;
12,7,4;
26,23,11,2;
59,71,41,8;
138,224,151,30;
		

Links

• David Callan, A bijection on Dyck paths and its cycle structure, El. J. Combinat. 14 (2007) # R28

Crossrefs

Cf. A127384 (row sums).

Programs

Maple

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 ;