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A241824
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.
5
0, 0, 1, 2, 4, 5, 9, 12, 19, 26, 38, 51, 72, 95, 129, 168, 223, 287, 374, 477, 613, 775, 984, 1234, 1552, 1932, 2408, 2978, 3684, 4527, 5563, 6797, 8301, 10090, 12253, 14821, 17910, 21564, 25936, 31099, 37245, 44482, 53064, 63142, 75046, 88994, 105406
OFFSET
0,4
FORMULA
a(n) = A241823(n) + A241824(n) for n >= 0.
a(n) + A241827(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 9 partitions: 42, 411, 33, 321, 3111, 222, 2211, 21111, 111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]
Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}] (* A241823 *)
Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)
Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)
Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)
Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}] (* A241827 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 30 2014
STATUS
approved