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%I #4 May 06 2014 15:05:46
%S 0,0,1,2,3,5,8,12,18,26,37,51,70,94,126,166,219,284,369,473,607,770,
%T 977,1228,1544,1925,2399,2970,3673,4517,5550,6784,8284,10073,12232,
%U 14799,17883,21536,25903,31064,37204,44439,53015,63090,74987,88932,105337
%N Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that min(x(i) - x(i-1)) < number of distinct parts of p.
%F a(n) = A241821(n) - A241820(n) for n >= 0.
%F a(n) + A241818(n) + A241820(n) = A000041(n) for n >= 0.
%e a(6) counts these 2 partitions: 51, 411
%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]
%t Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}] (* A241823 *)
%t Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)
%t Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)
%t Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)
%t Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}] (* A241827 *)
%Y Cf. A241824, A241825, A241826, A241827, A000041.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, Apr 30 2014
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