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A064327 Generalized Catalan numbers C(-5; n).

%I A064327 #15 Sep 08 2022 08:45:04
%S A064327 1,1,-4,41,-514,7206,-108174,1700721,-27646234,460887086,-7836596944,
%T A064327 135380098426,-2369445113804,41925242220616,-748729419265314,
%U A064327 13478117036893281,-244306305241572474,4455242518055441046,-81683397232911983784,1504758636166747742286
%N A064327 Generalized Catalan numbers C(-5; n).
%C A064327 See triangle A064334 with columns m built from C(-m; n), m >= 0, also for Derrida et al. references.
%H A064327 G. C. Greubel, <a href="/A064327/b064327.txt">Table of n, a(n) for n = 0..765</a>
%F A064327 a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-5)^m/n.
%F A064327 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).
%F A064327 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.
%F A064327 a(n) = hypergeometric([1-n, n], [-n], -5) for n > 0. - _Peter Luschny_, Nov 30 2014
%t A064327 CoefficientList[Series[(11 +Sqrt[1+20*x])/(2*(6-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)
%o A064327 (Sage)
%o A064327 def a(n):
%o A064327     if n==0: return 1
%o A064327     return hypergeometric([1-n, n], [-n], -5).simplify()
%o A064327 [a(n) for n in range(24)] # _Peter Luschny_, Nov 30 2014
%o A064327 (Sage) ((11 +sqrt(1+20*x))/(2*(6-x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 03 2019
%o A064327 (PARI) my(x='x+O('x^30)); Vec((11 +sqrt(1+20*x))/(2*(6-x))) \\ _G. C. Greubel_, May 03 2019
%o A064327 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (11 +Sqrt(1+20*x))/(2*(6-x)) )); // _G. C. Greubel_, May 03 2019
%Y A064327 Cf. A064334.
%K A064327 sign,easy
%O A064327 0,3
%A A064327 _Wolfdieter Lang_, Sep 21 2001