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%I #4 May 06 2014 15:06:44
%S 0,0,0,0,0,1,2,4,5,10,12,19,27,37,51,70,90,119,161,200,259,335,428,
%T 533,681,845,1065,1317,1636,2010,2496,3036,3719,4538,5523,6686,8096,
%U 9760,11759,14138,16943,20263,24216,28852,34308,40801,48315,57206,67590,79748
%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 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]]; g[p_] := Max[-Differences[p]];
%t Table[Count[f[n], p_ /; g[p] < d[p]], {n, 0, z}] (* A241818 *)
%t Table[Count[f[n], p_ /; g[p] <= d[p]], {n, 0, z}] (* A241819 *)
%t Table[Count[f[n], p_ /; g[p] == d[p]], {n, 0, z}] (* A241820 *)
%t Table[Count[f[n], p_ /; g[p] >= d[p]], {n, 0, z}] (* A241821 *)
%t Table[Count[f[n], p_ /; g[p] > d[p]], {n, 0, z}] (* A241822 *)
%Y Cf. A241818, A241819, A241820, A241821, A000041.
%K nonn,easy
%O 0,7
%A _Clark Kimberling_, Apr 30 2014