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A191525 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.

%I A191525 #10 Oct 20 2024 04:07:32
%S A191525 1,1,1,1,2,1,4,1,1,7,2,1,13,5,1,1,24,8,2,1,46,16,6,1,1,86,28,9,2,1,
%T A191525 166,58,19,7,1,1,314,103,32,10,2,1,610,211,71,22,8,1,1,1163,382,121,
%U A191525 36,11,2,1,2269,781,261,85,25,9,1,1,4352,1432,456,140,40,12,2,1,8518,2920,976,316,100,28,10,1,1
%N A191525 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n and having k hills; a hill is a (1,1)-step starting at level 0 and followed by a (1,-1)-step.
%C A191525 Row n contains 1+floor(n/2) entries.
%C A191525 Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C A191525 T(n,0)=A191526(n).
%C A191525 Sum(k*T(n,k), k>=0) = A037952(n-1).
%F A191525 G.f.: G(t,z)=1/((1-z*c)*(1-z^4*c^2-t*z^2)), where c=(1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.
%e A191525 T(5,1)=2 because we have (UD)UUD and (UD)UUU, where U=(1,1) and D=(1,-1) (the hills are shown between parentheses).
%e A191525 Triangle starts:
%e A191525   1;
%e A191525   1;
%e A191525   1,1;
%e A191525   2,1;
%e A191525   4,1,1;
%e A191525   7,2,1;
%e A191525   13,5,1,1;
%p A191525 c := ((1-sqrt(1-4*z^2))*1/2)/z^2: G := 1/((1-z*c)*(1-z^4*c^2-t*z^2)): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A191525 Cf. A001405, A191526, A037952, A065600.
%K A191525 nonn,tabf
%O A191525 0,5
%A A191525 _Emeric Deutsch_, Jun 06 2011