This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A064091 #25 Sep 08 2022 08:45:04
%S A064091 1,1,9,145,2905,65121,1563561,39322929,1022586105,27272680705,
%T A064091 741894295369,20504949587409,574176887116441,16254518495907745,
%U A064091 464436319229036265,13376293681432402545,387925710986712480825
%N A064091 Generalized Catalan numbers C(8; n).
%C A064091 a(n+1)= Y_{n}(n+1)= Z_{n}, n >= 0, in the Derrida et al. 1992 reference (see A064094) for alpha=8, beta =1 (or alpha=1, beta=8).
%H A064091 Vincenzo Librandi, <a href="/A064091/b064091.txt">Table of n, a(n) for n = 0..200</a>
%F A064091 G.f.: (1 + 8*x*c(8*x)/7)/(1+x/7) = 1/(1 - x*c(8*x)) with c(x) g.f. of Catalan numbers A000108.
%F A064091 a(n) = Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(8^m)/n.
%F A064091 a(n) = (-1/7)^n*(1 - 8*Sum_{k=0..n-1} C(k)*(-56)^k ), n >= 1, a(0) := 1; with C(n)=A000108(n) (Catalan).
%F A064091 a(n) = Sum_{k=0..n} A059365(n, k)*8^(n-k). - _Philippe Deléham_, Jan 19 2004
%F A064091 Conjecture: 7*n*a(n) +(-223*n+336)*a(n-1) +16*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jun 07 2013
%F A064091 a(n) ~ 2^(5*n+3)/(225*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Aug 13 2013
%t A064091 a[0]=1; a[n_]:= Sum[(n-m)*Binomial[n+m-1, m]*(8^m)/n, {m, 0, n-1}]; Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jun 21 2013 *)
%t A064091 Table[FullSimplify[(-1)^(2*n)*2^(3+5*n)*(1/2*(2*n-1))! Hypergeometric2F1[1,1/2+n,2+n,-224]/(Sqrt[Pi]*(n+1)!)],{n,0,20}] (* _Vaclav Kotesovec_, Aug 13 2013 *)
%t A064091 CoefficientList[Series[(15 -Sqrt[1-32*x])/(2*(x+7)), {x,0,20}], x] (* _G. C. Greubel_, May 02 2019 *)
%o A064091 (PARI) a(n)=if(n<0,0,polcoeff(serreverse((x-7*x^2)/(1+x)^2+O(x^(n+1))), n)) /* _Ralf Stephan_ */
%o A064091 (PARI) my(x='x+O('x^20)); Vec((15 -sqrt(1-32*x))/(2*(x+7))) \\ _G. C. Greubel_, May 02 2019
%o A064091 (Magma) R<x>:=PowerSeriesRing(Rationals(), 20); Coefficients(R!( (15 - Sqrt(1-32*x))/(2*(x+7)) )); // _G. C. Greubel_, May 02 2019
%o A064091 (Sage) ((15 -sqrt(1-32*x))/(2*(x+7))).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, May 02 2019
%Y A064091 Cf. A064090 (C(7, n)).
%Y A064091 Cf. A000108, A059365.
%K A064091 nonn,easy
%O A064091 0,3
%A A064091 _Wolfdieter Lang_, Sep 13 2001