A191791 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k UDUD's, where U=(1,1) and D=(1,-1).
1, 1, 2, 3, 5, 1, 8, 2, 15, 4, 1, 25, 8, 2, 46, 19, 4, 1, 79, 36, 9, 2, 147, 76, 24, 4, 1, 256, 146, 48, 10, 2, 477, 304, 109, 29, 4, 1, 841, 578, 224, 60, 11, 2, 1570, 1180, 499, 144, 34, 4, 1, 2791, 2244, 1002, 312, 72, 12, 2, 5217, 4525, 2172, 731, 181, 39, 4, 1, 9336, 8588, 4347, 1530, 410, 84, 13, 2
Offset: 0
Examples
T(7,2)=2 because we have (UD[UD)UD]U and U(UD[UD)UD], where U=(1,1) and D=(1,-1) (the UDUD's are shown between parentheses). Triangle starts: 1; 1; 2; 3; 5,1; 8,2; 15,4,1; 25,8,2;
Programs
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Maple
eq := z^2*(1+z^2-t*z^2)*C^2-(1+z^2+z^4-t*z^2-t*z^4)*C+1+z^2-t*z^2: C := RootOf(eq, C): G := C/(1-z*C): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)-1) end do; # yields sequence in triangular form
Formula
G.f.: G(t,z) = C/(1-z*C), where C=C(t,z) is given by z^2*(1+z^2-t*z^2)*C^2 - (1+z^2+z^4- t*z^2-t*z^4)*C + 1 + z^2 - t*z^2 = 0.
Comments