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 A182890 Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n. 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS 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. Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1). FORMULA G.f: x/((1+x+x^2)*(1-3*x+x^2)). a(n) = Sum_{k>=0} k*A182888(n,k). a(n) = (A000045(2n+2) - ((-1)^n)*A010892(n))/4. - John M. Campbell, Dec 30 2016 EXAMPLE a(3)=5. 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; they contain 0+0+1+1+3=5 h-steps at level 0. MAPLE 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); CROSSREFS Cf. A182888. Sequence in context: A244061 A297120 A299167 * A102714 A087223 A005955 Adjacent sequences:  A182887 A182888 A182889 * A182891 A182892 A182893 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Dec 12 2010 STATUS approved

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Last modified September 20 01:04 EDT 2020. Contains 337228 sequences. (Running on oeis4.)