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A064090 Generalized Catalan numbers C(7; n).

%I A064090 #26 Sep 08 2022 08:45:04
%S A064090 1,1,8,113,1982,38886,817062,17981769,409186310,9549411950,
%T A064090 227307541448,5497312072330,134696099554276,3336563455537768,
%U A064090 83419226227330722,2102274863070771033,53347639317495439302
%N A064090 Generalized Catalan numbers C(7; n).
%C A064090 a(n+1) = Y_{n}(n+1) = Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=7, beta =1 (or alpha=1, beta=7).
%H A064090 G. C. Greubel, <a href="/A064090/b064090.txt">Table of n, a(n) for n = 0..690</a>
%F A064090 G.f.: (1+7*x*c(7*x)/6)/(1+x/6) = 1/(1-x*c(7*x)) with c(x) g.f. of Catalan numbers A000108.
%F A064090 a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(7^m)/n.
%F A064090 a(n) = (-1/6)^n*(1 - 7*Sum_{k=0..n-1} C(k)*(-42)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F A064090 a(n) = Sum_{k=0..n} A059365(n, k)*7^(n-k). - _Philippe Deléham_, Jan 19 2004
%F A064090 Conjecture: 6*n*a(n) +(-167*n+252)*a(n-1) +14*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2013
%F A064090 a(n) ~ 4^n * 7^(n+1) / (169*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 10 2019
%t A064090 a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n-1+m, m]*7^m/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jul 09 2013 *)
%t A064090 CoefficientList[Series[(13 -Sqrt[1-28*x])/(2*(x+6)), {x, 0, 20}], x] (* _G. C. Greubel_, May 02 2019 *)
%o A064090 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-6*x^2)/(1+x)^2 +O(x^(n+1))), n)) /* _Ralf Stephan_ */
%o A064090 (PARI) my(x='x+O('x^20)); Vec((13-sqrt(1-28*x))/(2*(x+6))) \\ _G. C. Greubel_, May 02 2019
%o A064090 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (13-Sqrt(1-28*x))/(2*(x+6)) )); // _G. C. Greubel_, May 02 2019
%o A064090 (Sage) ((13-sqrt(1-28*x))/(2*(x+6))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 02 2019
%Y A064090 Cf. A064089 (C(6, n)).
%Y A064090 Cf. A000108, A059365.
%K A064090 nonn,easy
%O A064090 0,3
%A A064090 _Wolfdieter Lang_, Sep 13 2001