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%I #4 May 06 2014 15:06:28
%S 1,1,2,3,5,6,9,11,17,20,30,37,50,64,84,106,141,178,224,290,368,457,
%T 574,722,894,1113,1371,1693,2082,2555,3108,3806,4630,5605,6787,8197,
%U 9881,11877,14256,17047,20395,24320,28958,34409,40867,48333,57243,67548,79683
%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) = A241818(n) + A241820(n) for n >= 0.
%F a(n) + A241822(n) = A000041(n) for n >= 0.
%e a(6)= 9 counts all of the 11 partitions of 6 except 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, A241820, A241821, A241822, A000041.
%K nonn,easy
%O 0,3
%A _Clark Kimberling_, Apr 30 2014