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%I #5 May 06 2014 15:08:02
%S 0,0,0,0,0,1,1,3,3,6,7,12,14,22,27,37,46,63,76,101,124,160,196,250,
%T 302,382,463,574,693,855,1026,1255,1503,1823,2178,2626,3123,3749,4447,
%U 5305,6274,7458,8790,10405,12231,14422,16909,19871,23229,27217,31742
%N Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) > number of parts of p.
%F a(n) = A241831(n) - A241830(n).
%F a(n) + A241828(n) + A241830(n) = A000041(n) for n >= 0.
%e a(6) counts this single partition: 51.
%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]
%t Table[Count[f[n], p_ /; g[p] < Length[p]], {n, 0, z}] (* A241828 *)
%t Table[Count[f[n], p_ /; g[p] <= Length[p]], {n, 0, z}] (* A241829 *)
%t Table[Count[f[n], p_ /; g[p] == Length[p]], {n, 0, z}] (* A241830 *)
%t Table[Count[f[n], p_ /; g[p] >= Length[p]], {n, 0, z}] (* A241831 *)
%t Table[Count[f[n], p_ /; g[p] > Length[p]], {n, 0, z}] (* A241832 *)
%Y Cf. A241828, A241829, A241830, A241831, A000041.
%K nonn,easy
%O 0,8
%A _Clark Kimberling_, Apr 30 2014
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