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A239951 Number of partitions of n such that (number of distinct parts) > least part.

%I A239951 #10 Nov 17 2015 01:41:48
%S A239951 0,0,0,1,2,4,6,10,14,22,30,44,59,84,109,151,195,261,335,440,558,723,
%T A239951 909,1160,1452,1829,2272,2839,3503,4336,5326,6542,7984,9756,11842,
%U A239951 14376,17382,20985,25255,30355,36372,43528,51960,61925,73645,87460,103648,122650
%N A239951 Number of partitions of n such that (number of distinct parts) > least part.
%H A239951 Alois P. Heinz, <a href="/A239951/b239951.txt">Table of n, a(n) for n = 0..1000</a>
%F A239951 a(n) + A239949(n) = A000041(n) for n >= 0.
%e A239951 a(6) counts these 6 partitions:  51, 411, 321, 3111, 2211, 21111.
%p A239951 b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 0,
%p A239951       add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
%p A239951     end:
%p A239951 a:= n-> combinat[numbpart](n) -b(n$2, 0):
%p A239951 seq(a(n), n=0..80);  # _Alois P. Heinz_, Apr 02 2014
%t A239951 z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
%t A239951 Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
%t A239951 Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
%t A239951 Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
%t A239951 Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
%t A239951 Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
%t A239951 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_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 17 2015, after _Alois P. Heinz_ *)
%Y A239951 Cf. A239948, A239949, A239950, A239952.
%K A239951 nonn,easy
%O A239951 0,5
%A A239951 _Clark Kimberling_, Mar 30 2014