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A111133 Number of partitions of n into at least two distinct parts.

%I A111133 #70 Jun 23 2025 15:09:04
%S A111133 0,0,0,1,1,2,3,4,5,7,9,11,14,17,21,26,31,37,45,53,63,75,88,103,121,
%T A111133 141,164,191,221,255,295,339,389,447,511,584,667,759,863,981,1112,
%U A111133 1259,1425,1609,1815,2047,2303,2589,2909,3263,3657,4096,4581,5119,5717,6377
%N A111133 Number of partitions of n into at least two distinct parts.
%C A111133 Old name: Number of sets of natural numbers less than n which sum to n.
%C A111133 From _Clark Kimberling_, Mar 13 2012: (Start)
%C A111133 (1) Number of partitions of n into at least two distinct parts.
%C A111133 (2) Also, number of partitions of 2n into distinct parts having maximal part n; see Example section. (End)
%H A111133 Reinhard Zumkeller, <a href="/A111133/b111133.txt">Table of n, a(n) for n = 0..5000</a>
%H A111133 Riccardo Aragona, Roberto Civino, and Norberto Gavioli, <a href="https://arxiv.org/abs/2301.06347">A modular idealizer chain and unrefinability of partitions with repeated parts</a>, arXiv:2301.06347 [math.RA], 2023.
%H A111133 Thomas Enkosky and Branden Stone, <a href="http://arxiv.org/abs/1308.4945">Sequences defined by h-vectors</a>, arXiv:1308.4945 [math.CO], 2013.
%H A111133 Harmandeep Kaur and Muhammad Asif Rana, <a href="https://arxiv.org/abs/2506.11447">Partitions with unique largest part and their generating functions</a>, arXiv:2506.11447 [math.CO], 2025. See p. 6.
%F A111133 a(n) = A000009(n) - 1. - _Vladeta Jovovic_, Oct 19 2005
%F A111133 G.f.: Sum_{k>=0} (x^((k^2+k)/2) / Product_{j=1..k} (1-x^j)) - 1/(1-x). - _Joerg Arndt_, Sep 17 2012
%F A111133 a(n) = A026906(floor(n-1)/2) + A258259(n). - _Bob Selcoe_, Oct 05 2015
%F A111133 G.f.: -1/(1 - x) + Product_{k>=1} (1 + x^k). - _Ilya Gutkovskiy_, Aug 10 2018
%F A111133 G.f.: Sum_{n >= 1} x^(2*n+1)/Product_{k = 1..n+1} 1 - x^(2*k-1). - _Peter Bala_, Nov 20 2024
%e A111133 a(6) = 3 because 1+5, 2+4 and 1+2+3 each sum to 6. That is, the three sets are {1,5},{2,4},{1,2,3}.
%e A111133 For n=6, the partitions of 2n into distinct parts having maximum 6 are 6+5+1, 6+4+2, 6+3+2+1, so that a(6)=3, as an example for Comment (2). - _Clark Kimberling_, Mar 13 2012
%p A111133 seq(coeff(series(mul((1+x^k),k=1..n)-1/(1-x), x,n+1),x,n),n=0..60); # _Muniru A Asiru_, Aug 10 2018
%t A111133 Needs["DiscreteMath`Combinatorica`"]
%t A111133 f[n_] := Block[{lmt = Floor[(Sqrt[8n + 1] - 1)/2] + 1, t}, Sum[ Length[ Select[Plus @@@ KSubsets[ Range[n - k(k - 1)/2 + 1], k], # == n &]], {k, 2, lmt}]]; Array[f, 55] (* _Robert G. Wilson v_, Oct 17 2005 *)
%t A111133 (* Next program shows the partitions (sets) *)
%t A111133 d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@ #] == 1 &]; Table[d[n], {n, 1, 12}]
%t A111133 TableForm[%]
%t A111133 (* _Clark Kimberling_, Mar 13 2012 *)
%t A111133 Table[PartitionsQ[n]-1, {n, 0, 55}] (* _Jean-François Alcover_, Jan 17 2014, after _Vladeta Jovovic_ *)
%o A111133 (PARI) N=66;  x='x+O('x^N);
%o A111133 gf=sum(k=0,N, x^((k^2+k)/2) / prod(j=1,k, 1-x^j)) - 1/(1-x);
%o A111133 concat( [0,0,0], Vec(gf) ) /* _Joerg Arndt_, Sep 17 2012 */
%o A111133 (Haskell)
%o A111133 a111133 = subtract 1 . a000009  -- _Reinhard Zumkeller_, Sep 09 2015
%Y A111133 Cf. A058377.
%Y A111133 Cf. A000009.
%Y A111133 Cf. A026906, A258259.
%K A111133 nonn,nice
%O A111133 0,6
%A A111133 David Sharp (davidsharp(AT)rcn.com), Oct 17 2005
%E A111133 More terms from _Vladeta Jovovic_ and _Robert G. Wilson v_, Oct 17 2005
%E A111133 a(0)=0 prepended by _Joerg Arndt_, Sep 17 2012