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A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).

%I A227309 #33 Sep 05 2017 03:52:23
%S A227309 1,1,1,2,3,6,10,19,34,63,115,213,391,723,1333,2463,4547,8403,15522,
%T A227309 28686,53006,97963,181042,334606,618415,1142994,2112545,3904592,
%U A227309 7216810,13338856,24654268,45568784,84225393,155675230,287737327,531830605,982993368,1816887637,3358192905
%N A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).
%C A227309 Sums along falling diagonals of A161492 (skew Ferrers diagrams by area and number of columns). [_Joerg Arndt_, Mar 23 2014]
%H A227309 Vaclav Kotesovec, <a href="/A227309/b227309.txt">Table of n, a(n) for n = 0..1000</a>
%H A227309 M. P. Delest, J. M. Fedou, <a href="http://dx.doi.org/10.1016/0012-365X(93)90224-H">Enumeration of skew Ferrers diagrams</a>, Discrete Mathematics. vol.112, no.1-3, pp.65-79, (1993)
%F A227309 G.f.: 1/(1-q /(1-q^2/(1-q^2/(1-q^3/(1-q^3/(1-q^4/(1-q^4/(1-q^5/(1-q^5/(1-...) )) )) )) )) ).
%F A227309 G.f.: 1/x - Q(0)/(2*x), where Q(k)= 1 + 1/(1 - 1/(1 - 1/(2*x^(k+1)) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 09 2013
%F A227309 G.f.: 1/x - U(0)/x, where U(k)= 1 - x^(k+1)/(1 - x^(k+1)/U(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 15 2013
%F A227309 G.f.: -W(0)/x, where W(k)= 1 - x^(k+1) - x^k - x^(2*k+2)/W(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Aug 15 2013
%F A227309 G.f.: G(0) where G(k) = 1 - q/(q^(k+2) - 1 / G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jan 18 2016
%F A227309 a(n) ~ c * d^n, where d = 1.84832326133106924642685135202616091890310896530577301386219207630312784... and c = 0.244648950328338656997216931963422920467577616734159139510762093105072... - _Vaclav Kotesovec_, Sep 05 2017
%t A227309 nmax = 40; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[2, nmax] - Floor[Range[2, nmax]/2])]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 05 2017 *)
%o A227309 (PARI) N = 66;  q = 'q + O('q^N);
%o A227309 G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+2) / G(k+1) ) );
%o A227309 Vec( 1 / G(0) )
%o A227309 (PARI) /* formula from the Delest/Fedou reference with t=q: */
%o A227309 N=66;  q='q+O('q^N);  t=q;
%o A227309 qn(n) = prod(k=1, n, 1-q^k );
%o A227309 nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) );
%o A227309 dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n );
%o A227309 v=Vec(nm/dn)
%Y A227309 Cf. A049346 (g.f.: 1-1/G(0), G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
%Y A227309 Cf. A227310 (g.f.: 1/G(0), G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).
%Y A227309 Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A227309 Cf. A226729 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
%Y A227309 Cf. A006958 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
%K A227309 nonn
%O A227309 0,4
%A A227309 _Joerg Arndt_, Jul 06 2013