%I
%S 1,1,2,3,5,6,9,12,13,17,23,25,30,37,38,48,61,61,67,85,89,105,120,122,
%T 138,163,174,193,216,231,252,289,289,324,369,389,428,477,481,519,587,
%U 628,661,729,760,827,902,940,999,1095,1140,1238,1337,1380,1464,1616
%N Number of 'convex' partitions of n; i.e., partitions of n for which the Ferrers graph is the intersection of a convex set and the integer lattice.
%C Equivalently, a partition n = a_1 + ... + a_m with a_1 >= ... >= a_m >= 1 is convex if a_j >= floor(a_i + (a_k  a_i)*(ji)/(ki)) whenever 1 <= i < j < k <= m.
%C Can anyone supply a generating function or asymptotic formula?
%H Vaclav Kotesovec, <a href="/A074658/b074658.txt">Table of n, a(n) for n = 0..500</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram.</a>
%e The only partition of 5 which is not convex is 5 = 3 + 1 + 1, so a(5) = A000041(5)  1 = 6.
%t f[1, 1, 0, 0]=1; f[n_, m_, r_, s_] := Module[{nn, rr, ss}, If[GCD[r, s]!=12r*n>(m+1)(2r+s*m)(nn=nm*s+(r+1)(s1)/2)<mr, 0, f[n, m, r, s]=Sum[If[s*rr>=r*ss, f[nn, mr, rr, ss], 0], {rr, 0, mr1}, {ss, 0, nnm+r}]]]; a[n_] := Module[{r, s}, If[n<=1, 1, Sum[f[n, m, r, s], {m, 1, n}, {r, 0, m1}, {s, 0, nm}]]] (* f[n, m, r, s] = number of convex partitions of n into m parts, with bottom slope of convex hull equal to r/s *)
%Y The first and second differences are in A074659 and A074660.
%K nonn
%O 0,3
%A _Dean Hickerson_, Aug 29 2002
