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 A182895 Number of (1,0)-steps at level 0 in all weighted lattice paths in L_n. 3
 0, 1, 3, 7, 19, 50, 130, 341, 893, 2337, 6119, 16020, 41940, 109801, 287463, 752587, 1970299, 5158310, 13504630, 35355581, 92562113, 242330757, 634430159, 1660959720, 4348449000, 11384387281, 29804712843, 78029751247, 204284540899 (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 a(n) = Sum_{k>=0} k*A182893(n,k). G.f.: z(1+z)/[(1+z+z^2)(1-3z+z^2)]. a(n) = (A000032(2n+1) - A010892(2n))/4. - John M. Campbell, Dec 30 2016 EXAMPLE a(3) = 7. 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+2+2+3=7 (1,0)-steps at level 0. MAPLE G:=z*(1+z)/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G, z=0, 32): seq(coeff(Gser, z, n), n=0..28); CROSSREFS Cf. A182893. Sequence in context: A151266 A147234 A171854 * A087224 A308398 A341703 Adjacent sequences:  A182892 A182893 A182894 * A182896 A182897 A182898 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 12 2010 STATUS approved

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Last modified April 21 17:29 EDT 2021. Contains 343156 sequences. (Running on oeis4.)