cp's OEIS Frontend

A239953 Number of partitions of n such that twice the least part is the number of distinct parts.

%I A239953 #9 May 31 2021 05:37:15
%S A239953 0,0,0,1,2,4,5,8,9,12,14,17,18,23,25,28,33,39,44,54,61,77,92,112,131,
%T A239953 167,194,246,280,352,401,501,562,697,779,939,1055,1274,1401,1684,1846,
%U A239953 2186,2408,2825,3103,3617,3969,4583,5045,5801,6367,7304,8050,9150
%N A239953 Number of partitions of n such that twice the least part is the number of distinct parts.
%H A239953 Alois P. Heinz, <a href="/A239953/b239953.txt">Table of n, a(n) for n = 0..1000</a>
%e A239953 a(6) counts these 5 partitions :  51, 42, 3111, 22111, 21111.
%p A239953 b:= proc(n, i, d) option remember; `if`(2*min(i, n)<d+1, 0,
%p A239953       `if`(irem(n, i)=0 and 2*i=d+1, 1, b(n, i-1, d)+
%p A239953       add(b(n-i*j, i-1, d+1), j=1..n/i)))
%p A239953     end:
%p A239953 a:= n-> b(n$2, 0):
%p A239953 seq(a(n), n=0..60);  # _Alois P. Heinz_, Apr 02 2014
%t A239953 z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; Table[Count[ IntegerPartitions[n], p_ /; d[p] == 2 Min[p]], {n, 0, z}]  (* A239953 *)
%t A239953 (* Second program: *)
%t A239953 b[n_, i_, d_] := b[n, i, d] = If[2*Min[i, n] < d + 1, 0,
%t A239953      If[Mod[n, i] == 0 && 2*i == d + 1, 1, b[n, i - 1, d] +
%t A239953      Sum[b[n - i*j, i - 1, d + 1], {j, 1, n/i}]]];
%t A239953 a[n_] := b[n, n, 0];
%t A239953 a /@ Range[0, 60] (* _Jean-François Alcover_, May 31 2021, after _Alois P. Heinz_ *)
%Y A239953 Cf. A239948.
%K A239953 nonn,easy
%O A239953 0,5
%A A239953 _Clark Kimberling_, Mar 30 2014