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A064092 Generalized Catalan numbers C(9; n).

%I A064092 #21 Sep 08 2022 08:45:04
%S A064092 1,1,10,181,4078,102826,2777212,78571837,2298558934,68964092542,
%T A064092 2110472708140,65620725560578,2067160250751436,65833929303952564,
%U A064092 2116166898185821792,68565914052628406221,2237022199842087256678
%N A064092 Generalized Catalan numbers C(9; n).
%C A064092 a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=9, beta =1 (or alpha=1, beta=9).
%H A064092 G. C. Greubel, <a href="/A064092/b064092.txt">Table of n, a(n) for n = 0..640</a>
%F A064092 G.f.: (1 + 9*x*c(9*x)/8)/(1+x/8) = 1/(1 - x*c(9*x)) with c(x) g.f. of Catalan numbers A000108.
%F A064092 a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(9^m)/n.
%F A064092 a(n) = (-1/8)^n*(1 - 9*Sum_{k=0..n-1} C(k)*(-72)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F A064092 a(n) = Sum_{k=0..n} A059365(n, k)*9^(n-k). - _Philippe Deléham_, Jan 19 2004
%F A064092 Conjecture: 8*n*a(n) +(-287*n+432)*a(n-1) +18*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2013
%F A064092 a(n) ~ 4^n * 9^(n+1) / (289*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 10 2019
%t A064092 a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*9^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jul 09 2013 *)
%t A064092 CoefficientList[Series[(17 -Sqrt[1-36*x])/(2*(x+8)), {x, 0, 20}], x] (* _G. C. Greubel_, May 02 2019 *)
%o A064092 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-8*x^2)/(1+x)^2+O(x^(n+1))), n)) /* _Ralf Stephan_ */
%o A064092 (PARI) my(x='x+O('x^20)); Vec((17 -sqrt(1-36*x))/(2*(x+8))) \\ _G. C. Greubel_, May 02 2019
%o A064092 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (17 - Sqrt(1-36*x))/(2*(x+8)) )); // _G. C. Greubel_, May 02 2019
%o A064092 (Sage) ((17 -sqrt(1-36*x))/(2*(x+8))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 02 2019
%Y A064092 Cf. A064091 (C(8, n)).
%Y A064092 Cf. A000108, A059365.
%K A064092 nonn,easy
%O A064092 0,3
%A A064092 _Wolfdieter Lang_, Sep 13 2001