A111133 Number of partitions of n into at least two distinct parts.
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, 141, 164, 191, 221, 255, 295, 339, 389, 447, 511, 584, 667, 759, 863, 981, 1112, 1259, 1425, 1609, 1815, 2047, 2303, 2589, 2909, 3263, 3657, 4096, 4581, 5119, 5717, 6377
Offset: 0
Examples
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}.
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
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 0..5000
- Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
- Thomas Enkosky and Branden Stone, Sequences defined by h-vectors, arXiv:1308.4945 [math.CO], 2013.
- Harmandeep Kaur and Muhammad Asif Rana, Partitions with unique largest part and their generating functions, arXiv:2506.11447 [math.CO], 2025. See p. 6.
Programs
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Haskell
a111133 = subtract 1 . a000009 -- Reinhard Zumkeller, Sep 09 2015
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Maple
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
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Mathematica
Needs["DiscreteMath`Combinatorica`"] 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 *) (* Next program shows the partitions (sets) *) d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@ #] == 1 &]; Table[d[n], {n, 1, 12}] TableForm[%] (* Clark Kimberling, Mar 13 2012 *) Table[PartitionsQ[n]-1, {n, 0, 55}] (* Jean-François Alcover, Jan 17 2014, after Vladeta Jovovic *) -
PARI
N=66; x='x+O('x^N); gf=sum(k=0,N, x^((k^2+k)/2) / prod(j=1,k, 1-x^j)) - 1/(1-x); concat( [0,0,0], Vec(gf) ) /* Joerg Arndt, Sep 17 2012 */
Formula
a(n) = A000009(n) - 1. - Vladeta Jovovic, Oct 19 2005
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
G.f.: -1/(1 - x) + Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Aug 10 2018
G.f.: Sum_{n >= 1} x^(2*n+1)/Product_{k = 1..n+1} 1 - x^(2*k-1). - Peter Bala, Nov 20 2024
Extensions
More terms from Vladeta Jovovic and Robert G. Wilson v, Oct 17 2005
a(0)=0 prepended by Joerg Arndt, Sep 17 2012
Comments