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A064332
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Generalized Catalan numbers C(-10; n).
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3
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1, 1, -9, 181, -4529, 126861, -3806649, 119653941, -3889122369, 129646443421, -4408213959689, 152290162367301, -5330337257966609, 188617242067457581, -6736489341630231129, 242518500968942706261, -8791448318093732481249
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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)*(-10)^m/n.
a(n) = (1/11)^n*(1 + 10*Sum_{k=0..n-1} C(k)*(-10*11)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
G.f.: (1+10*x*c(-10*x)/11)/(1-x/11) = 1/(1-x*c(-10*x)) with c(x) g.f. of Catalan numbers A000108.
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MATHEMATICA
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CoefficientList[Series[(21 +Sqrt[1+40*x])/(2*(11-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((21 +sqrt(1+40*x))/(2*(11-x))) \\ G. C. Greubel, May 03 2019
(Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (21 +Sqrt(1+40*x))/(2*(11-x)) )); // G. C. Greubel, May 03 2019
(Sage) ((21 +sqrt(1+40*x))/(2*(11-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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