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%I #12 Sep 08 2022 08:45:04
%S 1,1,-6,85,-1490,29226,-614004,13511709,-307448490,7174776190,
%T -170777485556,4130050311234,-101192982385844,2506610481299380,
%U -62668163792277840,1579300030107459885,-40076101342241993370
%N Generalized Catalan numbers C(-7; 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="/A064329/b064329.txt">Table of n, a(n) for n = 0..690</a>
%F a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-7)^m/n.
%F 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).
%F 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.
%t CoefficientList[Series[(15 +Sqrt[1+28*x])/(2*(8-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((15 +sqrt(1+28*x))/(2*(8-x))) \\ _G. C. Greubel_, May 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (15 +Sqrt(1+28*x))/(2*(8-x)) )); // _G. C. Greubel_, May 03 2019
%o (Sage) ((15 +sqrt(1+28*x))/(2*(8-x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 03 2019
%K sign,easy
%O 0,3
%A _Wolfdieter Lang_, Sep 21 2001
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