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A015106
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Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-9.
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22
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1, 1, -8, -665, 483544, 3173511682, -187386353065808, -99585165693268026701, 476312561203989614441440600, 20503694883570579788445502041773422, -7943551457092331370323478258038812629918704
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-9 and a(0)=1.
G.f. satisfies: A(x) = 1 / (1 - x*A(-9*x)) = 1/(1-x/(1+9*x/(1-9^2*x/(1+9^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016
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EXAMPLE
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G.f. = 1 + x - 8*x^2 - 665*x^3 + 483544*x^4 + 3173511682*x^5 + ...
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MATHEMATICA
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m = 11; ContinuedFractionK[If[i == 1, 1, -(-9)^(i - 2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)
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PROG
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(Ruby)
def A(q, n)
ary = [1]
(1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
ary
end
A(-9, n)
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CROSSREFS
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Cf. A015108 (q=-11), A015107 (q=-10), this sequence (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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