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%I #11 Mar 31 2017 03:52:41
%S 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,
%T 31,47,83,71,19,1,49,80,132,140,55,6,109,162,223,263,140,26,1,170,292,
%U 377,468,316,86,7,371,592,693,830,665,246,34,1,581,1064,1264,1456,1314,622,126,8
%N 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).
%C Row n has 1 + ceiling(n/2) entries.
%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%C T(n,0) = A191393(n).
%C Sum_{k>=0} k*T(n,k) = A191394(n).
%H G. C. Greubel, <a href="/A191392/b191392.txt">Table of n, a(n) for the first 101 rows, flattened</a>
%F 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)).
%e 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).
%e Triangle starts:
%e 1;
%e 1;
%e 1, 1;
%e 1, 2;
%e 1, 4, 1;
%e 1, 6, 3;
%e 2, 9, 8, 1;
%e 3, 12, 16, 4;
%e 8, 18, 30, 13, 1;
%p 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
%t 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 *)
%Y Cf. A001405, A191393, A191394.
%K nonn,tabf
%O 0,6
%A _Emeric Deutsch_, Jun 04 2011
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