cp's OEIS Frontend

A240861 Number of partitions p of n into distinct parts not including the number of parts.

%I A240861 #24 Aug 11 2025 10:48:33
%S A240861 1,0,1,1,2,2,2,4,4,5,6,9,10,12,14,18,22,26,30,36,42,51,60,70,81,94,
%T A240861 110,128,148,172,198,226,260,298,342,390,446,508,577,654,742,840,951,
%U A240861 1074,1212,1366,1538,1728,1940,2176,2440,2732,3056,3416,3814,4254
%N A240861 Number of partitions p of n into distinct parts not including the number of parts.
%H A240861 Alois P. Heinz, <a href="/A240861/b240861.txt">Table of n, a(n) for n = 0..4000</a>
%H A240861 Atul Dixit, Gaurav Kumar, and Aviral Srivastava, <a href="https://arxiv.org/abs/2508.04359">Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions</a>, arXiv:2508.04359 [math.CO], 2025. See references.
%F A240861 a(n) = A000009(n) - A240855(n).
%e A240861 a(10) counts these 6 partitions:  {10}, {9,1}, {7,3}, {7,2,1}, {6,4}, {5,4,1}.
%p A240861 g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
%p A240861      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A240861     end:
%p A240861 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A240861      `if`(n=0, [x^p, 0], (f-> [add(coeff(f[1], x, j)*x^j
%p A240861         , j=i+1..degree(f[1])), f[2]+coeff(f[1], x, i)])(
%p A240861         b(n-i, min(n-i, i-1), p+1))+b(n, i-1, p)))
%p A240861     end:
%p A240861 a:= n-> g(n)-b(n$2, 0)[2]:
%p A240861 seq(a(n), n=0..55);  # _Alois P. Heinz_, Mar 14 2024
%t A240861 z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
%t A240861 Table[Count[f[n], p_ /; MemberQ[p, Length[p]]], {n, 0, z}]  (* A240855 *)
%t A240861 Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)
%Y A240861 Cf. A000009, A240855.
%K A240861 nonn,easy
%O A240861 0,5
%A A240861 _Clark Kimberling_, Apr 14 2014
%E A240861 a(0) changed to 1 by _Alois P. Heinz_, Mar 14 2024