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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.
5

%I #4 May 06 2014 15:07:55

%S 0,0,0,0,1,1,3,3,6,8,12,15,23,27,38,48,64,78,105,127,165,202,256,311,

%T 393,473,588,711,875,1050,1286,1537,1867,2229,2687,3195,3838,4544,

%U 5427,6416,7625,8981,10637,12492,14736,17269,20293,23715,27792,32391,37840

%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) = A241830(n) + A241832(n).

%F a(n) + A241828(n) = A000041(n) for n >= 0.

%e a(6) counts these 3 partitions: 51, 42, 411.

%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, A241832, A000041.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 30 2014