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A239949
Number of partitions of n such that (number of distinct parts) <= least part.
5
1, 1, 2, 2, 3, 3, 5, 5, 8, 8, 12, 12, 18, 17, 26, 25, 36, 36, 50, 50, 69, 69, 93, 95, 123, 129, 164, 171, 215, 229, 278, 300, 365, 387, 468, 507, 595, 652, 760, 830, 966, 1055, 1214, 1336, 1530, 1674, 1910, 2104, 2380, 2617, 2953, 3253, 3656, 4019, 4504
OFFSET
0,3
LINKS
FORMULA
a(n) + A239951(n) = A000041(n) for n >= 0.
EXAMPLE
a(8) counts these 8 partitions: 8, 62, 53, 44, 422, 332, 2222, 11111111.
MAPLE
b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 0,
add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80); # Alois P. Heinz, Apr 02 2014
MATHEMATICA
z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}] (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}] (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i <= d, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 30 2014
STATUS
approved