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 A182880 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,1)-steps. L_n is the set of lattice 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; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. 1
 1, 1, 2, 3, 2, 5, 6, 8, 18, 13, 44, 6, 21, 102, 30, 34, 222, 120, 55, 466, 390, 20, 89, 948, 1140, 140, 144, 1884, 3066, 700, 233, 3672, 7770, 2800, 70, 377, 7044, 18780, 9800, 630, 610, 13332, 43710, 31080, 3780, 987, 24946, 98610, 91560, 17850, 252, 1597, 46218, 216732, 254400, 72450, 2772 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Sum of entries in row n is A051286(n). REFERENCES 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. LINKS Table of n, a(n) for n=0..56. FORMULA T(n,0) = A000045(n+1) (the Fibonacci numbers). Sum_{k=0..n} k*T(n,k) = A182881(n). G.f.: G(t,z) = 1/sqrt(1 - 2*z - z^2 + 2*z^3 + z^4 - 4*t*z^3). The g.f. of column k is binomial(2n,n)*z^(3n)/(1-z-z^2)^(2n+1). Apparently, T(n,1) = 2*A001628(n-3), T(n,2) = 6*A001873(n-6), T(n,3) = 20*A001875(n-9). - R. J. Mathar, Dec 11 2010 EXAMPLE T(3,1)=2. 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; two of them, ud and du, have exactly one u step. Triangle starts: 1; 1; 2; 3, 2; 5, 6; 8, 18; 13, 44, 6; MAPLE G:=1/sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^3): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 16 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 16 do seq(coeff(P[n], t, k), k=0..floor(n/3)) od; # yields sequence in triangular form CROSSREFS Cf. A051286, A000045, A182881. Sequence in context: A066729 A241592 A211506 * A182898 A133684 A286151 Adjacent sequences: A182877 A182878 A182879 * A182881 A182882 A182883 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 11 2010 STATUS approved

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Last modified February 26 17:57 EST 2024. Contains 370352 sequences. (Running on oeis4.)