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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 15:07:27

%S 1,1,2,3,5,6,10,12,19,24,35,44,63,79,108,139,185,234,309,389,503,632,

%T 806,1005,1273,1576,1973,2436,3025,3710,4578,5587,6846,8320,10132,

%U 12257,14854,17888,21568,25880,31064,37125,44384,52856,62944,74712,88649,104883

%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) = A241828(n) + A241830(n).

%F a(n) + A241832(n) = A000041(n) for n >= 0.

%e a(6) counts these 10 partitions: 6, 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.

%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, A241830, A241831, A241832, A000041.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 30 2014