A059777 Number of self-conjugate three-quadrant Ferrers graphs that partition n.

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, 228, 273, 329, 393, 465, 551, 655, 776, 911, 1070, 1261, 1480, 1726, 2014, 2354, 2742, 3180, 3688, 4279, 4954, 5716, 6590, 7603, 8754, 10049, 11532

Offset: 0

Author

N. J. A. Sloane, Feb 21 2001

Keywords

• nonn

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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

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

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

a(n) ~ exp(sqrt(n/2)*Pi)/(8*sqrt(2)*n). - Vaclav Kotesovec, Sep 26 2016

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