A191392 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0)-steps at positive heights) having k base pyramids. A base pyramid is a factor of the form U^j D^j (j > 0), starting at the horizontal axis. Here U = (1,1) and D = (1,-1).
1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 2, 9, 8, 1, 3, 12, 16, 4, 8, 18, 30, 13, 1, 13, 26, 50, 32, 5, 31, 47, 83, 71, 19, 1, 49, 80, 132, 140, 55, 6, 109, 162, 223, 263, 140, 26, 1, 170, 292, 377, 468, 316, 86, 7, 371, 592, 693, 830, 665, 246, 34, 1, 581, 1064, 1264, 1456, 1314, 622, 126, 8
Offset: 0
Examples
T(5,2)=3 because we have H(UD)(UD), (UD)H(UD), and (UD)(UD)H, where U=(1,1), D=(1,-1), H=(1,0) (the base pyramids are shown between parentheses). Triangle starts: 1; 1; 1, 1; 1, 2; 1, 4, 1; 1, 6, 3; 2, 9, 8, 1; 3, 12, 16, 4; 8, 18, 30, 13, 1;
Links
- G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Programs
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Maple
G := (2*(1-z^2))/(1-2*z+z^2+2*z^3-2*t*z^2+(1-z^2)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
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Mathematica
CoefficientList[CoefficientList[Series[(2*(1 - z^2))/(1 - 2*z + z^2 + 2*z^3 - 2*t*z^2 + (1 - z^2)*Sqrt[1 - 4*z^2]), {z, 0, 10}, {t, 0, 10}], z], t] // Flatten (* G. C. Greubel, Mar 29 2017 *)
Formula
G.f.: G(t,z) = (2*(1 - z^2))/(1 - 2*z + z^2 + 2*z^3 - 2*t*z^2 + (1 - z^2)*sqrt(1 - 4*z^2)).
Comments