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A241818
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.
6
1, 1, 2, 3, 4, 5, 7, 9, 12, 17, 20, 27, 37, 46, 59, 81, 102, 130, 170, 212, 273, 344, 432, 539, 679, 843, 1049, 1297, 1602, 1968, 2422, 2961, 3608, 4395, 5334, 6467, 7800, 9418, 11311, 13593, 16287, 19482, 23214, 27702, 32908, 39117, 46305, 54856, 64749
OFFSET
0,3
COMMENTS
For the partition [n] of n, "max(x(i) - x(i-1))" is (as in the Mathematica program) interpreted as 0.
FORMULA
a(n) + A241820(n) + A241821(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 7 partitions: 6, 33, 321, 222, 2211, 21111, 111111.
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