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A241822
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 distinct parts of p.
5
0, 0, 0, 0, 0, 1, 2, 4, 5, 10, 12, 19, 27, 37, 51, 70, 90, 119, 161, 200, 259, 335, 428, 533, 681, 845, 1065, 1317, 1636, 2010, 2496, 3036, 3719, 4538, 5523, 6686, 8096, 9760, 11759, 14138, 16943, 20263, 24216, 28852, 34308, 40801, 48315, 57206, 67590, 79748
OFFSET
0,7
FORMULA
a(n) = A241821(n) - A241820(n) for n >= 0.
a(n) + A241818(n) + A241820(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 2 partitions: 51, 411.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g[p_] := Max[-Differences[p]];
Table[Count[f[n], p_ /; g[p] < d[p]], {n, 0, z}] (* A241818 *)
Table[Count[f[n], p_ /; g[p] <= d[p]], {n, 0, z}] (* A241819 *)
Table[Count[f[n], p_ /; g[p] == d[p]], {n, 0, z}] (* A241820 *)
Table[Count[f[n], p_ /; g[p] >= d[p]], {n, 0, z}] (* A241821 *)
Table[Count[f[n], p_ /; g[p] > d[p]], {n, 0, z}] (* A241822 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 30 2014
STATUS
approved