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A239949 Number of partitions of n such that (number of distinct parts) <= least part.

%I A239949 #10 Nov 17 2015 01:35:30
%S A239949 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,
%T A239949 129,164,171,215,229,278,300,365,387,468,507,595,652,760,830,966,1055,
%U A239949 1214,1336,1530,1674,1910,2104,2380,2617,2953,3253,3656,4019,4504
%N A239949 Number of partitions of n such that (number of distinct parts) <= least part.
%H A239949 Alois P. Heinz, <a href="/A239949/b239949.txt">Table of n, a(n) for n = 0..1000</a>
%F A239949 a(n) + A239951(n) = A000041(n) for n >= 0.
%e A239949 a(8) counts these 8 partitions:  8, 62, 53, 44, 422, 332, 2222, 11111111.
%p A239949 b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 0,
%p A239949       add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
%p A239949     end:
%p A239949 a:= n-> b(n$2, 0):
%p A239949 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 02 2014
%t A239949 z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
%t A239949 Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
%t A239949 Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
%t A239949 Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
%t A239949 Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
%t A239949 Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
%t A239949 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_ *)
%Y A239949 Cf. A239948, A239950, A239951, A239952.
%K A239949 nonn,easy
%O A239949 0,3
%A A239949 _Clark Kimberling_, Mar 30 2014