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A182878
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Triangle read by rows: T(n,k) is the number of lattice paths L_n of weight n having length k (0 <= k <= n). These are paths that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1.
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4
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1, 0, 1, 0, 1, 1, 0, 0, 4, 1, 0, 0, 1, 9, 1, 0, 0, 0, 9, 16, 1, 0, 0, 0, 1, 36, 25, 1, 0, 0, 0, 0, 16, 100, 36, 1, 0, 0, 0, 0, 1, 100, 225, 49, 1, 0, 0, 0, 0, 0, 25, 400, 441, 64, 1, 0, 0, 0, 0, 0, 1, 225, 1225, 784, 81, 1, 0, 0, 0, 0, 0, 0, 36, 1225, 3136, 1296, 100, 1, 0, 0, 0, 0, 0, 0, 1, 441, 4900, 7056, 2025, 121, 1
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OFFSET
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0,9
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COMMENTS
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The weight of a path is the sum of the weights of its steps.
Sum of entries in row n is A051286(n).
Sum_{k=0..n} k*T(n,k) = A182879(n).
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REFERENCES
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M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
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LINKS
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FORMULA
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T(n,k) = binomial(n,n-k)^2.
G.f. = G(t,z) = ((1-t*z)^2 - 2*t*z^2 - 2*t^2*z^3 + t^2*z^4)^(-1/2).
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EXAMPLE
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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 hhh, hH, Hh, ud, and du, having lengths 3, 2, 2, 2, and 2, respectively.
Triangle starts:
1;
0, 1;
0, 1, 1;
0, 0, 4, 1;
0, 0, 1, 9, 1;
0, 0, 0, 9, 16, 1;
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MAPLE
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T:=(n, k)->binomial(k, n-k)^2: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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