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A241828 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
1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 30, 41, 54, 74, 97, 128, 167, 219, 280, 363, 462, 590, 746, 944, 1182, 1485, 1848, 2299, 2843, 3515, 4318, 5305, 6482, 7914, 9623, 11688, 14139, 17093, 20588, 24769, 29713, 35602, 42537, 50769, 60439, 71865, 85265, 101039 (list; graph; refs; listen; history; text; internal format)
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.
LINKS
FORMULA
a(n) = A241826(n) - A241825(n).
a(n) + A241823(n) + A241825(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 8 partitions: 6, 33, 321, 3111, 222, 2211, 21111, 111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]
Table[Count[f[n], p_ /; g[p] < Length[p]], {n, 0, z}] (* A241828 *)
Table[Count[f[n], p_ /; g[p] <= Length[p]], {n, 0, z}] (* A241829 *)
Table[Count[f[n], p_ /; g[p] == Length[p]], {n, 0, z}] (* A241830 *)
Table[Count[f[n], p_ /; g[p] >= Length[p]], {n, 0, z}] (* A241831 *)
Table[Count[f[n], p_ /; g[p] > Length[p]], {n, 0, z}] (* A241832 *)
CROSSREFS
Sequence in context: A180652 A046682 A005987 * A125895 A241344 A064428
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 30 2014
STATUS
approved

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Last modified April 23 03:30 EDT 2024. Contains 371906 sequences. (Running on oeis4.)