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%I #16 Nov 06 2019 01:12:09
%S 1,1,2,4,1,8,3,17,9,37,25,1,82,66,5,185,171,20,423,437,70,1,978,1107,
%T 225,7,2283,2790,686,35,5373,7009,2015,147,1,12735,17574,5760,553,9,
%U 30372,44019,16135,1932,54,72832,110210,44500,6398,264,1,175502,275925,121247,20350,1134,11
%N Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,-1)-returns to the horizontal axis. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
%H Andrew Howroyd, <a href="/A182896/b182896.txt">Table of n, a(n) for n = 0..1750</a>
%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.
%H E. Munarini, N. Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.
%F G.f.: G(t,z) = 1/(1-z-z^2-(1+t)z^3*c), where c satisfies c = 1 + zc + z^2*c + z^3*c^2.
%F Sum of entries in row n is A051286(n).
%F T(n,0) = A004148(n+1) (the secondary structure numbers).
%F Sum_{k=0..n} k*T(n,k) = A182897(n).
%e T(3,1)=1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; exactly one of them, namely ud, has one (1,-1)-return to the horizontal axis.
%e Triangle starts:
%e 1;
%e 1;
%e 2;
%e 4, 1;
%e 8, 3;
%e 17, 9;
%e 37, 25, 1;
%e 82, 66, 5;
%e ...
%p eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-t*z^3*c-z^3*c): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do: #yields sequence in triangular form
%o (PARI)
%o T(n)={[Vecrev(p) | p<-Vec(1/(1-x-x^2 - (1+y)*(1-x-x^2 - sqrt(1+x^4-2*x^3-x^2-2*x+O(x*x^n)))/2))]}
%o { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Nov 05 2019
%Y Cf. A051286, A004148, A182897.
%K nonn,walk,tabf
%O 0,3
%A _Emeric Deutsch_, Dec 12 2010
%E Data corrected by _Andrew Howroyd_, Nov 05 2019
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