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A064329
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Generalized Catalan numbers C(-7; n).
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4
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1, 1, -6, 85, -1490, 29226, -614004, 13511709, -307448490, 7174776190, -170777485556, 4130050311234, -101192982385844, 2506610481299380, -62668163792277840, 1579300030107459885, -40076101342241993370
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OFFSET
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0,3
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COMMENTS
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See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
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LINKS
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FORMULA
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a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-7)^m/n.
a(n) = (1/8)^n*(1 + 7*Sum_{k=0..n-1} C(k)*(-7*8)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+7*x*c(-7*x)/8)/(1-x/8) = 1/(1-x*c(-7*x)) with c(x) g.f. of Catalan numbers A000108.
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MATHEMATICA
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CoefficientList[Series[(15 +Sqrt[1+28*x])/(2*(8-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((15 +sqrt(1+28*x))/(2*(8-x))) \\ G. C. Greubel, May 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (15 +Sqrt(1+28*x))/(2*(8-x)) )); // G. C. Greubel, May 03 2019
(Sage) ((15 +sqrt(1+28*x))/(2*(8-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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