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%I #12 Sep 08 2022 08:45:04
%S 1,1,-9,181,-4529,126861,-3806649,119653941,-3889122369,129646443421,
%T -4408213959689,152290162367301,-5330337257966609,188617242067457581,
%U -6736489341630231129,242518500968942706261,-8791448318093732481249
%N Generalized Catalan numbers C(-10; n).
%C See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
%H G. C. Greubel, <a href="/A064332/b064332.txt">Table of n, a(n) for n = 0..625</a>
%F a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-10)^m/n.
%F 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).
%F 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.
%t CoefficientList[Series[(21 +Sqrt[1+40*x])/(2*(11-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((21 +sqrt(1+40*x))/(2*(11-x))) \\ _G. C. Greubel_, May 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (21 +Sqrt(1+40*x))/(2*(11-x)) )); // _G. C. Greubel_, May 03 2019
%o (Sage) ((21 +sqrt(1+40*x))/(2*(11-x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 03 2019
%Y Cf. A000108, A064334.
%K sign,easy
%O 0,3
%A _Wolfdieter Lang_, Sep 21 2001
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