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A182890
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Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n.
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3
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0, 1, 2, 5, 14, 36, 94, 247, 646, 1691, 4428, 11592, 30348, 79453, 208010, 544577, 1425722, 3732588, 9772042, 25583539, 66978574, 175352183, 459077976, 1201881744, 3146567256, 8237820025, 21566892818, 56462858429, 147821682470, 387002188980, 1013184884470
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OFFSET
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0,3
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COMMENTS
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The members of L_n are paths of weight n 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. The weight of a path is the sum of the weights of its steps.
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LINKS
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FORMULA
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G.f: x/((1+x+x^2)*(1-3*x+x^2)).
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EXAMPLE
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a(3)=5. Indeed, denoting by h (resp. H) the (1,0)-step of weight 1 (resp. 2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 0+0+1+1+3=5 h-steps at level 0.
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MAPLE
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G:=z/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G, z=0, 33): seq(coeff(Gser, z, n), n=0..30);
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MATHEMATICA
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Table[Sum[Binomial[2n+2-2k, 2k-1]/2, {k, 0, n+1}], {n, 0, 30}]; (* Rigoberto Florez, Apr 10 2023 *)
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PROG
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(Maxima) a(n):=1/2*sum(binomial(2*n-2*m, 2*m+1), m, 0, (2*n-1)/4); /* Vladimir Kruchinin, Jan 24 2022 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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