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

0,5

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

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

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 *)

Crossrefs

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

Sequence in context: A285799 A241772 A323053 * A017830 A274149 A026928

Adjacent sequences: A059774 A059775 A059776 * A059778 A059779 A059780

Keyword

nonn

Author

N. J. A. Sloane, Feb 21 2001

Status

approved