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%I #15 Sep 08 2022 08:45:04
%S 1,1,-3,25,-251,2817,-33843,425769,-5537835,73865617,-1004862179,
%T 13888533561,-194475377243,2752994728225,-39333541106835,
%U 566464908534345,-8214515461250955,119845125957958065,-1757855400878129475,25906894146115000665,-383443906519878272955
%N Generalized Catalan numbers C(-4; 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="/A064326/b064326.txt">Table of n, a(n) for n = 0..830</a>
%F a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-4)^m/n.
%F a(n) = (1/5)^n*(1 + 4*Sum_{k=0..n-1} C(k)*(-4*5)^k), n >= 1, a(0) = 1; with C(n) = A000108(n) (Catalan).
%F G.f.: (1+4*x*c(-4*x)/5)/(1-x/5) = 1/(1-x*c(-4*x)) with c(x) g.f. of Catalan numbers A000108.
%F a(n) = hypergeometric([1-n, n], [-n], -4) for n > 0. - _Peter Luschny_, Nov 30 2014
%t CoefficientList[Series[(9 +Sqrt[1+16*x])/(2*(5-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)
%o (Sage)
%o def a(n):
%o if n==0: return 1
%o return hypergeometric([1-n, n], [-n], -4).simplify()
%o [a(n) for n in range(20)] # _Peter Luschny_, Nov 30 2014
%o (Sage) ((9 +sqrt(1+16*x))/(2*(5-x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 03 2019
%o (PARI) my(x='x+O('x^30)); Vec((9 +sqrt(1+16*x))/(2*(5-x))) \\ _G. C. Greubel_, May 03 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9 +Sqrt(1+16*x))/(2*(5-x)) )); // _G. C. Greubel_, May 03 2019
%Y Cf. A064334.
%K sign,easy
%O 0,3
%A _Wolfdieter Lang_, Sep 21 2001
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