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 A182896 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,-1)-returns to the horizontal axis. The members of L_n are paths of weight n that start at (0,0), end on 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. 3
 1, 1, 2, 4, 1, 8, 3, 17, 9, 37, 25, 1, 82, 66, 5, 185, 171, 20, 423, 437, 70, 1, 978, 1107, 225, 7, 2283, 2790, 686, 35, 5373, 7009, 2015, 147, 1, 12735, 17574, 5760, 553, 9, 30372, 44019, 16135, 1932, 54, 72832, 110210, 44500, 6398, 264, 1, 175502, 275925, 121247, 20350, 1134, 11 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1750 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. FORMULA G.f.: G(t,z) = 1/(1-z-z^2-(1+t)z^3*c), where c satisfies c = 1 + zc + z^2*c + z^3*c^2. Sum of entries in row n is A051286(n). T(n,0) = A004148(n+1) (the secondary structure numbers). Sum_{k=0..n} k*T(n,k) = A182897(n). EXAMPLE T(3,1)=1. 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; exactly one of them, namely ud, has one (1,-1)-return to the horizontal axis. Triangle starts:    1;    1;    2;    4,  1;    8,  3;   17,  9;   37, 25, 1;   82, 66, 5;   ... MAPLE eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-t*z^3*c-z^3*c): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do: #yields sequence in triangular form PROG (PARI) T(n)={[Vecrev(p) | p<-Vec(1/(1-x-x^2 - (1+y)*(1-x-x^2 - sqrt(1+x^4-2*x^3-x^2-2*x+O(x*x^n)))/2))]} { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 05 2019 CROSSREFS Cf. A051286, A004148, A182897. Sequence in context: A232723 A275486 A065278 * A207605 A112931 A121685 Adjacent sequences:  A182893 A182894 A182895 * A182897 A182898 A182899 KEYWORD nonn,walk,tabf AUTHOR Emeric Deutsch, Dec 12 2010 EXTENSIONS Data corrected by Andrew Howroyd, Nov 05 2019 STATUS approved

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Last modified April 21 17:36 EDT 2021. Contains 343156 sequences. (Running on oeis4.)