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
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 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) / 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
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
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