A059777 - OEIS

%I #38 Jul 10 2022 15:35:57

%S 1,0,1,1,2,2,3,4,6,7,9,12,16,19,24,31,39,47,58,72,89,107,129,158,192,

%T 228,273,329,393,465,551,655,776,911,1070,1261,1480,1726,2014,2354,

%U 2742,3180,3688,4279,4954,5716,6590,7603,8754,10049,11532

%N Number of self-conjugate three-quadrant Ferrers graphs that partition n.

%D G. E. Andrews, Three-quadrant Ferrers graphs, Indian J. Math., 42 (No. 1, 2000), 1-7.

%H Seiichi Manyama, <a href="/A059777/b059777.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: 1/((1+x)*Sum_{k>=0} (-x)^(k*(k+1)/2)). [Corrected by _N. J. A. Sloane_, Jul 10 2022 at the suggestion of Eduardo Brietzke.] a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*(A002129(k)-1)*a(n-k). A006950(n) = a(n-1) + a(n), n > 0. - _Vladeta Jovovic_, Sep 22 2002

%F G.f.: 1/((1+x)*G(0)), where G(k)= 1 - x^(2*k+1)/(1 - x^(2*k+2)/(x^(2*k+2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 20 2013

%F G.f.: conjecture: 1/(Q(0) - 1), where Q(k) = 1 + (-x)^k - (-x)^(k+2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 25 2013

%F a(n) ~ exp(sqrt(n/2)*Pi)/(8*sqrt(2)*n). - _Vaclav Kotesovec_, Sep 26 2016

%F G.f.: Sum_{k>=0} x^(2*k) * Product_{j=1..k} (1+x^(2*j-1))/(1-x^(2*j)). - _Seiichi Manyama_, Jul 11 2018

%p mul((1+q^(2*n+3))/(1-q^(2*n+2)), n=0..101); # g.f.

%t nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k + 1))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 26 2016 *)

%Y Cf. A002129, A006950, A059776, A059778, A316675.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Feb 21 2001