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Number of partitions of n such that (number of distinct parts) > least part.
5

%I #10 Nov 17 2015 01:41:48

%S 0,0,0,1,2,4,6,10,14,22,30,44,59,84,109,151,195,261,335,440,558,723,

%T 909,1160,1452,1829,2272,2839,3503,4336,5326,6542,7984,9756,11842,

%U 14376,17382,20985,25255,30355,36372,43528,51960,61925,73645,87460,103648,122650

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

%H Alois P. Heinz, <a href="/A239951/b239951.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) + A239949(n) = A000041(n) for n >= 0.

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

%p b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 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..80); # _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, 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 Cf. A239948, A239949, A239950, A239952.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Mar 30 2014