A227309 G.f.: 1/G(0) where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ).
1, 1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 213, 391, 723, 1333, 2463, 4547, 8403, 15522, 28686, 53006, 97963, 181042, 334606, 618415, 1142994, 2112545, 3904592, 7216810, 13338856, 24654268, 45568784, 84225393, 155675230, 287737327, 531830605, 982993368, 1816887637, 3358192905
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
- M. P. Delest, J. M. Fedou, Enumeration of skew Ferrers diagrams, Discrete Mathematics. vol.112, no.1-3, pp.65-79, (1993)
Crossrefs
Cf. A049346 (g.f.: 1-1/G(0), G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Cf. A227310 (g.f.: 1/G(0), G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ) ).
Cf. A226728 (g.f.: 1/G(0), G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A226729 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).
Cf. A006958 (g.f.: 1/G(0), G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).
Programs
-
Mathematica
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 *) -
PARI
N = 66; q = 'q + O('q^N); G(k) = if(k>N, 1, 1 - q^(k+1) / (1 - q^(k+2) / G(k+1) ) ); Vec( 1 / G(0) ) -
PARI
/* formula from the Delest/Fedou reference with t=q: */ N=66; q='q+O('q^N); t=q; qn(n) = prod(k=1, n, 1-q^k ); nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) ); dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n ); v=Vec(nm/dn)
Formula
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-...) )) )) )) )) ).
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
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
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
G.f.: G(0) where G(k) = 1 - q/(q^(k+2) - 1 / G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2016
a(n) ~ c * d^n, where d = 1.84832326133106924642685135202616091890310896530577301386219207630312784... and c = 0.244648950328338656997216931963422920467577616734159139510762093105072... - Vaclav Kotesovec, Sep 05 2017
Comments