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%I #23 Mar 16 2024 09:11:00
%S 1,1,5,65,1725,81225,6181125,710984625,117537778125,26848583825625,
%T 8210318193703125,3275053250628290625,1667519951972905828125,
%U 1063947235962694359515625,837322677987349287566953125,801714108831393845941434140625
%N G.f.: 1/(1-x/(1-4*x/(1-10*x/(1-20*x/(1-35*x/(1-56*x/(1-...))))))), a continued fraction.
%H Vaclav Kotesovec, <a href="/A218221/b218221.txt">Table of n, a(n) for n = 0..198</a>
%F a(n) == 0 (mod 5) for n>1.
%F a(n) == 0 (mod 3) for n>3.
%F G.f.: Q(0), where Q(k) = 1 - x*(k+1)*(k+2)*(k+3)/(x*(k+1)*(k+2)*(k+3) - 6/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 19 2013
%F a(n) ~ c * d^n * (n!)^3 * n^(5/2), where d = 32*sqrt(3)*Pi^3 / Gamma(1/3)^9 = 0.241821816937867322064737317131437... and c = 2^(15/2) * 3^(11/4) * Pi^4 / Gamma(1/3)^(27/2) = 0.60382458700655692263505976... - _Vaclav Kotesovec_, Aug 25 2017, updated Mar 16 2024
%e G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1725*x^4 + 81225*x^5 +...
%t nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Binomial[Range[nmax + 1]+2,3]*x]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2017 *)
%t eulerCF[f_, len_] := Module[{g}, g[len - 1] = 1;
%t g[k_] := g[k] = 1 - f[k]/(f[k] - 2/g[k + 1]); CoefficientList[g[0] + O[x]^len, x]];
%t A218221List[len_] := eulerCF[(1/3) x (# + 1) (# + 2) (# + 3) &, len];
%t A218221List[16] (* _Peter Luschny_, Aug 09 2019 *)
%o (PARI) {a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)*(n-k+2)*(n-k+3)/6*x*CF)); polcoeff(CF, n)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A000292, A002105, A216966.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 23 2012
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