%I #17 Nov 17 2015 01:32:01

%S 0,1,1,2,3,6,8,13,18,26,36,50,68,92,123,162,214,279,360,464,595,754,

%T 959,1206,1513,1893,2358,2918,3615,4451,5462,6691,8174,9940,12081,

%U 14631,17675,21314,25637,30763,36861,44059,52555,62600,74417,88287,104600,123716

%N Number of partitions of n such that (number of distinct parts) >= least part.

%H Alois P. Heinz, <a href="/A239952/b239952.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) + A239948(n) = A000041(n) for n >= 0.

%e a(6) counts these 8 partitions: 51, 42, 411, 321, 3111, 2211, 21111, 111111.

%p b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d+1, 0,

%p add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))

%p end:

%p a:= n-> combinat[numbpart](n) -b(n$2, 0):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Apr 02 2014

%t z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];

%t Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}] (*A239948*)

%t Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)

%t Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)

%t Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}] (*A239951*)

%t Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)

%t b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i<=d+1, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Nov 17 2015, after _Alois P. Heinz_ *)

%Y Cf. A239948, A239949, A239950, A239951.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, Mar 30 2014