%I #23 Feb 11 2023 12:04:13
%S 1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,
%T 11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,18,18,18,19,
%U 19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,25,25,25
%N Number of monomial ideals in two variables x, y that are Artinian, integrally closed, of colength n and contain x^2.
%C Alternatively, "concave partitions" of n with at most 2 parts, where a concave partition is defined by demanding that the monomial ideal, generated by the monomials whose exponents do not lie in the Ferrers diagram of the partition, is integrally closed.
%D G. E. Andrews, The Theory of Partitions, Addison-Wesley Publishing Company, 1976.
%D M. Paulsen and J. Snellman, Enumerativa egenskaper hos konkava partitioner (in Swedish), Department of Mathematics, Stockholm University.
%H V. Crispin Quinonez, <a href="https://www2.math.su.se/reports/2002/7/2002-7.pdf">Integrally closed monomial ideals and powers of ideals</a>, Research Reports in Mathematics Number 7 2002, Department of Mathematics, Stockholm University.
%H Jan Snellman and Michael Paulsen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Snellman/snellman2.html">Enumeration of Concave Integer Partitions</a>, J. Integer Seqs., Vol. 7, 2004.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F G.f.: (1 + x^2 - x^3)/((1 - x)*(1 - x^3)).
%F a(n) = A008620(n+1). - _R. J. Mathar_, Sep 12 2008
%F E.g.f.: (3*exp(x)*(3 + x) - 2*sqrt(3)*exp(-x/2)*sin(sqrt(3)*x/2))/9. - _Stefano Spezia_, Feb 11 2023
%o (PARI) Vec((1+x^2-x^3)/((1-x)*(1-x^3)) + O(x^80)) \\ _Michel Marcus_, May 22 2015
%Y Cf. A008620, A084913, A086162, A086163.
%K nonn,easy
%O 0,3
%A Jan Snellman (Jan.Snellman(AT)math.su.se), Aug 25 2003