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%I #6 Dec 02 2014 16:28:11
%S 1,0,0,0,0,1,0,0,0,1,0,0,0,0,2,0,0,0,0,1,3,0,0,0,0,1,2,5,0,0,0,0,1,2,
%T 6,8,0,0,0,0,1,2,8,13,13,0,0,0,0,1,2,10,19,29,21,0,0,0,0,1,2,12,25,51,
%U 60,34,0,0,0,0,1,2,14,31,78,120,122,55,0,0,0,0,1,2,16,37,110,200,282,241
%N Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having pyramid weight k.
%C A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a Dyck word 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 Dyck path (word) is the sum of the heights of its maximal pyramids.
%C Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
%C T(n,n)=A000045(n-1) (n>=1; the Fibonacci numbers).
%C Sum(k*T(n,k), k>=0)=A166302(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 G.f.: G=G(t,z) satisfies G = 1 + zG[G - 1 + tz - tz(1 - t)/(1 - tz)].
%e T(6,5)=2 because we have U(UUDD)(UUUDDD)D and U(UUUDDD)(UUDD)D (the maximal pyramids are shown between parentheses).
%e Triangle starts:
%e 1;
%e 0,0;
%e 0,0,1;
%e 0,0,0,1;
%e 0,0,0,0,2;
%e 0,0,0,0,1,3;
%e 0,0,0,0,1,2,5;
%e 0,0,0,0,1,2,6,8;
%p eq := G = 1+z*G*(G-1+t^2*z*(1-z)/(1-t*z)): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 12 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
%Y Cf. A004148, A166302, A000045, A091866.
%K nonn,tabl
%O 0,15
%A _Emeric Deutsch_, Nov 07 2009
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