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A000782
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a(n) = 2*Catalan(n) - Catalan(n-1).
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6
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1, 3, 8, 23, 70, 222, 726, 2431, 8294, 28730, 100776, 357238, 1277788, 4605980, 16715250, 61020495, 223931910, 825632610, 3056887680, 11360977650, 42368413620, 158498860260, 594636663660, 2236748680998, 8433988655580, 31872759742852, 120699748759856
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OFFSET
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1,2
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COMMENTS
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Number of Dyck (n+1)-paths that have a leading or trailing hill. - David Scambler, Aug 22 2012
a(n) is the number of parking functions of size n avoiding the patterns 132, 213, 312, and 321. - Lara Pudwell, Apr 10 2023
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LINKS
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FORMULA
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Expansion of x*(1 + x*C)*C^2, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for the Catalan numbers, A000108.
Also, expansion of (1 + x^2*C^2)*C - 1, where C = (1 - (1 - 4*x)^(1/2))/(2*x) is the g.f. for Catalan numbers, A000108.
a(n) = leftmost column term of M^(n-1)*V, where M is a tridiagonal matrix with 1's in the super- and subdiagonals, (1, 2, 2, 2, ...) in the main diagonal, and the rest zeros; and V is the vector [1, 2, 0, 0, 0, ...]. - Gary W. Adamson, Jun 16 2011
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MATHEMATICA
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CoefficientList[Series[(1+x*(1-(1-4*x)^(1/2))/(2*x)^1)*((1-(1-4*x)^(1/2))/(2*x))^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 10 2012 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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