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%I #17 Jul 17 2017 02:16:15
%S 1,1,1,1,1,2,1,3,2,1,4,5,1,5,10,4,1,6,16,12,1,7,24,30,8,1,8,33,56,28,
%T 1,9,44,98,84,16,1,10,56,152,179,64,1,11,70,228,358,224,32,1,12,85,
%U 320,618,536,144,1,13,102,440,1030,1206,576,64,1,14,120,580,1580,2292,1528,320,1,15,140,754,2370,4202,3820,1440,128
%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 pyramid weight equal to k.
%C A pyramid in a dispersed Dyck path is a factor of the form U^h D^h, h being the height of the pyramid and U=(1,1), D=(1,-1). A pyramid in a dispersed Dyck path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a dispersed Dyck path is the sum of the heights of its maximal pyramids.
%C Row n has 1 + floor(n/2) entries.
%C Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
%H A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.
%F T(n,0) = 1;
%F T(n,1) = n-1 (n>=1).
%F T(n,2) = A001859(n-3) (n>=4).
%F Sum_{k>=0} k*T(n,k) = A191319(n).
%F G.f.: G=G(t,z) satisfies z*(1-z)*(z-1+2*t*z^2)*G^2 + (1-z)*(z-1+2*t*z^2)*G+1-t*z^2=0.
%e T(6,2)=10 because we have HH(UD)(UD), HH(UUDD), H(UD)H(UD), H(UD)(UD)H, H(UUDD)H, (UD)HH(UD), (UD)H(UD)H, (UD)(UD)HH, (UUDD)HH, and U(UD)(UD)D, where U=(1,1), D=(1,-1), H=(1,0); the maximal pyramids are shown between parentheses.
%e Triangle starts:
%e 1;
%e 1;
%e 1, 1;
%e 1, 2;
%e 1, 3, 2;
%e 1, 4, 5;
%e 1, 5, 10, 4;
%e 1, 6, 16, 12;
%e 1, 7, 24, 30, 8;
%p a := (z-1)*(2*t*z^2+z-1): c := -1+t*z^2: eq := a*z*G^2+a*G+c: f := RootOf(eq, G): fser := simplify(series(f, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(fser, 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 Cf. A001405, A001859, A191319.
%K nonn,tabf
%O 0,6
%A _Emeric Deutsch_, Jun 01 2011
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