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Generalized Catalan numbers C(-8; n).
3

%I #10 Sep 08 2022 08:45:04

%S 1,1,-7,113,-2263,50721,-1217703,30622929,-796311415,21237226625,

%T -577699502407,15966537989425,-447086291268119,12656524451911393,

%U -361628025405250023,10415207118205622673,-302049007052246016183

%N Generalized Catalan numbers C(-8; 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="/A064330/b064330.txt">Table of n, a(n) for n = 0..660</a>

%F a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(-8)^m/n.

%F a(n) = (1/9)^n*(1 + 8*Sum_{k=0..n-1} C(k)*(-8*9)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).

%F G.f.: (1+8*x*c(-8*x)/9)/(1-x/9) = 1/(1-x*c(-8*x)) with c(x) g.f. of Catalan numbers A000108.

%t CoefficientList[Series[(17 +Sqrt[1+32*x])/(2*(9-x)), {x, 0, 30}], x] (* _G. C. Greubel_, May 03 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((17 +sqrt(1+32*x))/(2*(9-x))) \\ _G. C. Greubel_, May 03 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (17 +Sqrt(1+32*x))/(2*(9-x)) )); // _G. C. Greubel_, May 03 2019

%o (Sage) ((17 +sqrt(1+32*x))/(2*(9-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