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A182883
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Number of weighted lattice paths of weight n having no (1,0)-steps of weight 1.
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4
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1, 0, 1, 2, 1, 6, 7, 12, 31, 40, 91, 170, 281, 602, 1051, 1988, 3907, 7044, 13735, 25962, 48643, 94094, 177145, 338184, 647791, 1228812, 2356927, 4500678, 8595913, 16486966, 31521543, 60419872, 115870879, 222045160, 426275647, 818054654
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OFFSET
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0,4
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COMMENTS
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L_n is the set of lattice 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; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
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REFERENCES
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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.
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LINKS
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FORMULA
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G.f.: 1/sqrt(1-2*z^2+z^4-4*z^3).
It appears that a(n) = Sum_{k = 0..floor(n/2)} binomial(n-k,k)*binomial(k,n-2*k): this gives correct values for a(0) through a(35). If true, then sequence equals antidiagonal sums of triangle A105868. - Peter Bala, Mar 06 2013
D-finite n*a(n) = (2*n - 2)*a(n-2) + (4*n - 6)*a(n-3) - (n - 2)*a(n-4), follows easily by differentiating the o.g.f. Maple's sumrecursion command verifies that Sum_{k = 0..floor(n/2)} binomial(n-k,k)*binomial(k,n-2*k) satisfies the same recurrence with the same initial conditions thus proving the above conjecture. - Peter Bala, Feb 07 2017
a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*hypergeom([-k, k-n, k-n], [1, -n], -1). - Peter Luschny, Feb 13 2018
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EXAMPLE
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a(3)=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, namely ud and du, contain no h steps.
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MAPLE
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G:=1/sqrt(1-2*z^2+z^4-4*z^3): Gser:=series(G, z=0, 40): seq(coeff(Gser, z, n), n=0..35);
# Alternatively (after Bala):
seq(add(binomial(n-k, k)*binomial(k, n-2*k), k=ceil(n/3)...floor(n/2)), n=0..35); # Peter Luschny, Feb 07 2017
# With natural summation bound:
a := n -> add((-1)^k*binomial(n, k)*hypergeom([-k, k-n, k-n], [1, -n], -1), k=0..n): seq(simplify(a(n)), n=0..35); # Peter Luschny, Feb 13 2018
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MATHEMATICA
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CoefficientList[Series[1/Sqrt[1-2x^2+x^4-4x^3], {x, 0, 40}], x] (* Harvey P. Dale, Oct 16 2011 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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