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A064088 Generalized Catalan numbers C(5; n).

%I A064088 #32 Jan 07 2025 19:05:44
%S A064088 1,1,6,61,766,10746,161376,2537781,41260086,687927166,11698135396,
%T A064088 202104763026,3537486504556,62595852983236,1117926476207316,
%U A064088 20124876291104421,364797768048805926,6652740911381353206,121975721251036497636,2247064873245590484966,41573071647518070152196
%N A064088 Generalized Catalan numbers C(5; n).
%C A064088 a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=5, beta =1 (or alpha=1, beta=5).
%H A064088 G. C. Greubel, <a href="/A064088/b064088.txt">Table of n, a(n) for n = 0..850</a>
%F A064088 G.f.: (1+5*x*c(5*x)/4)/(1+x/4) = 1/(1-x*c(5*x)) with c(x) g.f. of Catalan numbers A000108.
%F A064088 a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(5^m)/n, n >= 1, a(0) := 1.
%F A064088 a(n) = (-1/4)^n*(1 - 5*Sum_{k=0..n-1} C(k)*(-20)^k); with C(n)=A000108(n) (Catalan).
%F A064088 a(n) = Sum_{k=0..n} A059365(n, k)*5^(n-k). - _Philippe Deléham_, Jan 19 2004
%F A064088 From _Gary W. Adamson_, Jul 18 2011: (Start)
%F A064088 a(n) = upper left term in M^n, M = an infinite square production matrix as follows:
%F A064088   1, 1, 0, 0, 0, 0, ...
%F A064088   5, 5, 5, 0, 0, 0, ...
%F A064088   5, 5, 5, 5, 0, 0, ...
%F A064088   5, 5, 5, 5, 5, 0, ...
%F A064088   5, 5, 5, 5, 5, 5, ...
%F A064088   ... (End)
%F A064088 Conjecture: 4*n*a(n) +(-79*n+120)*a(n-1) +10*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2013
%F A064088 a(n) ~ 4^n * 5^(n+1) / (81*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 10 2019
%t A064088 a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*5^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 17}] (* _Jean-François Alcover_, Jul 09 2013 *)
%t A064088 CoefficientList[Series[(9-Sqrt[1-20*x])/(2*(x+4)), {x,0,30}], x] (* _G. C. Greubel_, May 02 2019 *)
%o A064088 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-4*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* _Ralf Stephan_ */
%o A064088 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (9-Sqrt(1-20*x))/(2*(x+4)) )); // _G. C. Greubel_, May 02 2019
%o A064088 (Sage) ( (9-sqrt(1-20*x))/(2*(x+4)) ).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 02 2019
%Y A064088 Cf. A064087 (C(4, n)).
%K A064088 nonn,easy
%O A064088 0,3
%A A064088 _Wolfdieter Lang_, Sep 13 2001