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A064327
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Generalized Catalan numbers C(-5; n).
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4
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1, 1, -4, 41, -514, 7206, -108174, 1700721, -27646234, 460887086, -7836596944, 135380098426, -2369445113804, 41925242220616, -748729419265314, 13478117036893281, -244306305241572474, 4455242518055441046, -81683397232911983784, 1504758636166747742286
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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)*(-5)^m/n.
a(n) = (1/6)^n*(1 + 5*Sum_{k=0..n-1} C(k)*(-5*6)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+5*x*c(-5*x)/6)/(1-x/6) = 1/(1-x*c(-5*x)) with c(x) g.f. of Catalan numbers A000108.
a(n) = hypergeometric([1-n, n], [-n], -5) for n > 0. - Peter Luschny, Nov 30 2014
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MATHEMATICA
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CoefficientList[Series[(11 +Sqrt[1+20*x])/(2*(6-x)), {x, 0, 30}], x] (* G. C. Greubel, May 03 2019 *)
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PROG
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(Sage)
def a(n):
if n==0: return 1
return hypergeometric([1-n, n], [-n], -5).simplify()
(Sage) ((11 +sqrt(1+20*x))/(2*(6-x))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 03 2019
(PARI) my(x='x+O('x^30)); Vec((11 +sqrt(1+20*x))/(2*(6-x))) \\ G. C. Greubel, May 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (11 +Sqrt(1+20*x))/(2*(6-x)) )); // 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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