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A064089 Generalized Catalan numbers C(6; n).

%I A064089 #20 Sep 08 2022 08:45:04
%S A064089 1,1,7,85,1279,21517,387607,7312789,142648495,2853691357,58226571271,
%T A064089 1207062556261,25351452769567,538285926177325,11535690316148215,
%U A064089 249189167966657845,5420206822556721295
%N A064089 Generalized Catalan numbers C(6; n).
%C A064089 a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=6, beta =1 (or alpha=1, beta=6).
%H A064089 G. C. Greubel, <a href="/A064089/b064089.txt">Table of n, a(n) for n = 0..725</a>
%F A064089 G.f.: (1 + 6*x*c(6*x)/5)/(1+x/5) = 1/(1 - x*c(6*x)) with c(x) g.f. of Catalan numbers A000108.
%F A064089 a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(6^m)/n.
%F A064089 a(n) = (-1/5)^n*(1 - 6*Sum_{k=0..n-1} C(k)*(-30)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F A064089 a(n) = Sum_{k=0..n} A059365(n, k)*6^(n-k). - _Philippe Deléham_, Jan 19 2004
%F A064089 Conjecture: 5*n*a(n) +(-119*n+180)*a(n-1) +12*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2013
%F A064089 a(n) ~ 2^(3*n + 1) * 3^(n+1) / (121*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 10 2019
%t A064089 CoefficientList[Series[(11 -Sqrt[1-24*x])/(2*(x+5)), {x, 0, 20}], x] (* _G. C. Greubel_, May 02 2019 *)
%o A064089 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-5*x^2)/(1+x)^2+O(x^(n+1))), n)) /* _Ralf Stephan_ */
%o A064089 (PARI) my(x='x+O('x^20)); Vec((11 -sqrt(1-24*x))/(2*(x+5))) \\ _G. C. Greubel_, May 02 2019
%o A064089 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (11 -Sqrt(1-24*x))/(2*(x+5)) )); // _G. C. Greubel_, May 02 2019
%o A064089 (Sage) ((11 -sqrt(1-24*x))/(2*(x+5))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 02 2019
%Y A064089 Cf. A064088 (C(5, n)).
%Y A064089 Cf. A000108, A059365.
%K A064089 nonn,easy
%O A064089 0,3
%A A064089 _Wolfdieter Lang_, Sep 13 2001