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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 15:07:20

%S 1,1,2,3,4,6,8,12,16,22,30,41,54,74,97,128,167,219,280,363,462,590,

%T 746,944,1182,1485,1848,2299,2843,3515,4318,5305,6482,7914,9623,11688,

%U 14139,17093,20588,24769,29713,35602,42537,50769,60439,71865,85265,101039

%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.

%C For the partition [n] of n, "max(x(i) - x(i-1))" is (as in the Mathematica program) interpreted as 0.

%F a(n) = A241826(n) - A241825(n).

%F a(n) + A241823(n) + A241825(n) = A000041(n) for n >= 0.

%e a(6) counts these 8 partitions: 6, 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. A241829, A241830, A241831, A241832, A000041.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Apr 30 2014