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A239948 Number of partitions of n such that (number of distinct parts) < least part.

%I A239948 #8 Oct 12 2015 04:04:35
%S A239948 1,0,1,1,2,1,3,2,4,4,6,6,9,9,12,14,17,18,25,26,32,38,43,49,62,65,78,
%T A239948 92,103,114,142,151,175,203,229,252,302,323,378,422,477,524,619,661,
%U A239948 758,847,958,1038,1204,1297,1485,1626,1829,1989,2285,2459,2770,3035
%N A239948 Number of partitions of n such that (number of distinct parts) < least part.
%H A239948 Alois P. Heinz, <a href="/A239948/b239948.txt">Table of n, a(n) for n = 0..1000</a>
%F A239948 a(n) + A239952(n) = A000041(n) for n >= 0.
%e A239948 a(10) counts these 6 partitions:  [10], [7,3], [6,4], [5,5], [4,3,3], [2,2,2,2,2].
%p A239948 b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d+1, 0,
%p A239948       add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
%p A239948     end:
%p A239948 a:= n-> b(n$2, 0):
%p A239948 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 02 2014
%t A239948 z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
%t A239948 Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
%t A239948 Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
%t A239948 Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
%t A239948 Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
%t A239948 Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
%t A239948 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_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Oct 12 2015, after _Alois P. Heinz_ *)
%Y A239948 Cf. A239949, A239950, A239951, A239952.
%K A239948 nonn,easy
%O A239948 0,5
%A A239948 _Clark Kimberling_, Mar 30 2014