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 A111133 Number of partitions of n into at least two distinct parts. 16
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Old name: Number of sets of natural numbers less than n which sum to n. From Clark Kimberling, Mar 13 2012: (Start) (1) Number of partitions of n into at least two distinct parts. (2) Also, number of partitions of 2n into distinct parts having maximal part n; see Example section. (End) LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 T. Enkosky, B. Stone, Sequences defined by h-vectors, arXiv preprint arXiv:1308.4945 [math.CO], 2013 FORMULA a(n) = A000009(n) - 1. - Vladeta Jovovic, Oct 19 2005 G.f. sum(k>=0, x^((k^2+k)/2) / prod(j=1..k, 1-x^j)) - 1/(1-x). - Joerg Arndt, Sep 17 2012 a(n) = A026906(floor(n-1)/2)) + A258259(n). - Bob Selcoe, Oct 05 2015 G.f.: -1/(1 - x) + Product_{k>=1} (1 + x^k). - Ilya Gutkovskiy, Aug 10 2018 EXAMPLE 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 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 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 *) PROG (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 */ (Haskell) a111133 = subtract 1 . a000009  -- Reinhard Zumkeller, Sep 09 2015 CROSSREFS Cf. A058377. Cf. A000009. Cf. A026906, A258259. Sequence in context: A161306 A266744 A242216 * A279076 A076970 A064548 Adjacent sequences:  A111130 A111131 A111132 * A111134 A111135 A111136 KEYWORD nonn,nice AUTHOR David Sharp (davidsharp(AT)rcn.com), Oct 17 2005 EXTENSIONS More terms from Vladeta Jovovic and Robert G. Wilson v, Oct 17 2005 a(0)=0 prepended by Joerg Arndt, Sep 17 2012 STATUS approved

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Last modified October 19 04:40 EDT 2019. Contains 328211 sequences. (Running on oeis4.)