A241824 - OEIS

%I #4 May 06 2014 15:06:57

%S 0,0,1,2,4,5,9,12,19,26,38,51,72,95,129,168,223,287,374,477,613,775,

%T 984,1234,1552,1932,2408,2978,3684,4527,5563,6797,8301,10090,12253,

%U 14821,17910,21564,25936,31099,37245,44482,53064,63142,75046,88994,105406

%N Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >=  ... >= x(k), of n such that min(x(i) - x(i-1)) <= number of distinct parts of p.

%F a(n) = A241823(n) + A241824(n) for n >= 0.

%F a(n) + A241827(n) = A000041(n) for n >= 0.

%e a(6) counts these 9 partitions:  42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]

%t Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}]  (* A241823 *)

%t Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)

%t Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)

%t Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)

%t Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}]  (* A241827 *)

%Y Cf. A241823, A241825, A241826, A241827, A000041.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Apr 30 2014