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%I #13 Sep 08 2022 08:45:04
%S 1,1,-5,61,-917,15421,-277733,5239117,-102188021,2044131037,
%T -41706059525,864547613293,-18157111255829,385517710342909,
%U -8261602828082213,178459989617336461,-3881680470161846837
%N Generalized Catalan numbers C(-6; 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="/A064328/b064328.txt">Table of n, a(n) for n = 0..720</a>
%F a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-6)^m/n.
%F a(n) = (1/7)^n*(1 + 6*Sum_{k=0..n-1} C(k)*(-6*7)^k), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F G.f.: (1+6*x*c(-6*x)/7)/(1-x/7) = 1/(1-x*c(-6*x)) with c(x) g.f. of Catalan numbers A000108.
%t CoefficientList[Series[(13 +Sqrt[1+24*x])/(2*(7-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec((13 +sqrt(1+24*x))/(2*(7-x))) \\ _G. C. Greubel_, May 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (13 +Sqrt(1+24*x))/(2*(7-x)) )); // _G. C. Greubel_, May 03 2019
%o (Sage) ((13 +sqrt(1+24*x))/(2*(7-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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