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 A182902 Number of valleys in all weighted lattice paths in B(n). 1
 0, 0, 0, 0, 0, 0, 1, 4, 14, 45, 135, 391, 1105, 3067, 8404, 22806, 61428, 164495, 438459, 1164363, 3082717, 8141422, 21457255, 56455195, 148323305, 389213825, 1020283146, 2672225692, 6993600748, 18291536552, 47814575243, 124929304664, 326280023426 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The members of B(n) are paths of weight n that start at (0,0), end on but never go below 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. A valley is a (1,-1)-step followed by a (1,1)-step. REFERENCES M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306. LINKS Table of n, a(n) for n=0..32. FORMULA a(n) = Sum(k*A182900(n,k), k>=0). G.f.: G:=z^6*g^4/(1-z^3*g^2), where g=g(z) satisfies g=1+zg+z^2*g+z^3*g^2. Conjecture D-finite with recurrence -3*(n+3)*(n-6)*a(n) +(n+1)*(7*n-34)*a(n-1) +2*(5*n+26)*a(n-2) +(7*n^2-39*n+16)*a(n-3) +4*(-n^2+5*n+2)*a(n-4) +(3*n^2-29*n+64)*a(n-5) -(n-4)*(n-7)*a(n-6)=0. - R. J. Mathar, Jul 22 2022 EXAMPLE a(7) = 4. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U = (1,1), D = (1,-1), among the 82 paths in B(7) only hUDUD, UDUDh, UDUhD, and UhDUD have valleys (1 in each). MAPLE eq := g = 1+z*g+z^2*g+z^3*g^2: g := RootOf(eq, g): gser := series(z^6*g^4/(1-z^3*g^2), z = 0, 35): seq(coeff(gser, z, n), n = 0 .. 32); CROSSREFS A182900. Sequence in context: A125068 A184138 A318019 * A108765 A304068 A005775 Adjacent sequences: A182899 A182900 A182901 * A182903 A182904 A182905 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 15 2010 STATUS approved

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Last modified June 9 03:24 EDT 2023. Contains 363168 sequences. (Running on oeis4.)