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%I #5 Sep 17 2014 20:53:05
%S 1,1,2,3,1,5,2,1,8,5,3,1,13,11,8,4,1,21,24,19,12,5,1,34,52,44,31,17,6,
%T 1,55,112,101,76,48,23,7,1,89,241,230,183,125,71,30,8,1,144,518,524,
%U 436,315,197,101,38,9,1,233,1113,1193,1037,781,520,299,139,47,10,1
%N Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k weak peaks.
%C B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.
%C A weak peak in a lattice path is a vertex on the top of a hump. A hump is an upstep followed by 0 or more flatsteps followed by a downstep. For example, the weighted lattice path Hu*duu*h*H*dd has 4 weak peaks (shown by the stars).
%C Row n contains n-1 entries (n>=2).
%C Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).
%C T(n,0) = A000045(n+1) (the Fibonacci numbers).
%C Sum(k*T(n,k), k=0..n) = A247470(n).
%H M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F G.f. G = G(t,z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - 1/(1 - z - z^2) + t/(1 - t*z - t*z^2)).
%e Row 3 is 3, 1 because B(3) = {u*d, hH, Hh, hhh} (weak peaks shown by *).
%e Triangle starts:
%e 1;
%e 1;
%e 2;
%e 3,1;
%e 5,2,1;
%e 8,5,3,1;
%e 13,11,8,4,1;
%p eq := G = 1+z*G+z^2*G+z^3*G*(G-1/(1-z-z^2)+t/(1-t*z-t*z^2)): G := RootOf(eq, G): 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 .. n-2) end do; # yields sequence in triangular form
%Y Cf. A000045, A004148, A247470,
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Sep 17 2014
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