A026906 -id:A026906 - cp's OEIS frontend

A056870 Difference between partial sums of partition numbers (A026905) and partial sums of numbers of partitions into distinct parts (A026906).

0, 1, 2, 5, 9, 16, 26, 42, 64, 96, 140, 202, 285, 398, 547, 746, 1005, 1344, 1780, 2343, 3059, 3972, 5123, 6576, 8392, 10663, 13481, 16977, 21286, 26594, 33096, 41055, 50750, 62548, 76846, 94155, 115032, 140183, 170386, 206611, 249934, 301682

Offset: 1

Robert G. Wilson v, Sep 02 2000

nonn

Cf. A026905, A026906, A056871.

Table[ Sum[ PartitionsP[k], {k, 1, n}] - Sum[PartitionsQ[k], {k, 1, n}], {n, 1, 45}]
(* second program *)
Module[{nn=50},#[[1]]-#[[2]]&/@Thread[{Accumulate[PartitionsP[Range[nn]]],Accumulate[PartitionsQ[Range[nn]]]}]] (* Harvey P. Dale, Jun 19 2025 *)

A056871 Sum of partial sums of partition numbers (A026905) and partial sums of numbers of partitions into distinct parts (A026906).

Original entry on oeis.org

2, 5, 10, 17, 27, 42, 62, 90, 128, 180, 248, 340, 459, 616, 819, 1082, 1417, 1848, 2392, 3083, 3951, 5042, 6401, 8098, 10198, 12799, 16001, 19941, 24762, 30662, 37844, 46583, 57174, 69996, 85464, 104109, 126506, 153385, 185552, 224003, 269846

Offset: 1

Robert G. Wilson v, Sep 02 2000

nonn

Table[ Sum[ PartitionsP[k], {k, 1, n}] + Sum[PartitionsQ[k], {k, 1, n}], {n, 1, 45}]

Cf. A026905, A026906, A056870.

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

Original entry on oeis.org

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

David Sharp (davidsharp(AT)rcn.com), Oct 17 2005

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)

			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

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.

Cf. A058377.
Cf. A000009.
Cf. A026906, A258259.

a111133 = subtract 1 . a000009  -- Reinhard Zumkeller, Sep 09 2015

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

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 *)

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 */

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
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
G.f.: Sum_{n >= 1} x^(2*n+1)/Product_{k = 1..n+1} 1 - x^(2*k-1). - Peter Bala, Nov 20 2024

More terms from Vladeta Jovovic and Robert G. Wilson v, Oct 17 2005

a(0)=0 prepended by Joerg Arndt, Sep 17 2012

A026905 Partial sums of the partition numbers A000041 of the positive integers.

Original entry on oeis.org

1, 3, 6, 11, 18, 29, 44, 66, 96, 138, 194, 271, 372, 507, 683, 914, 1211, 1596, 2086, 2713, 3505, 4507, 5762, 7337, 9295, 11731, 14741, 18459, 23024, 28628, 35470, 43819, 53962, 66272, 81155, 99132, 120769, 146784, 177969, 215307, 259890, 313064, 376325, 451500

Offset: 1

Clark Kimberling

nonn

Equivalently, a(n) = number of sums S of positive integers satisfying S <= n.
Equivalently, first differences give A000041. - Jacques ALARDET, Aug 04 2008, Aug 15 2008
For the partial sums of the partitions numbers of nonnegative integers A001477 see A000070. - Omar E. Pol, Nov 12 2011
Also number of parts in all regions of n that contain 1 as a part (Cf. A206437). - Omar E. Pol, Mar 11 2012
Also the number of graph minors of the path graph P_n (not counting the null graph). - Eric W. Weisstein, Apr 29 2022

Riccardo Aragona, Roberto Civino, and Norberto Gavioli, An ultimately periodic chain in the integral Lie ring of partitions, J. Algebr. Comb. (2024). See p. 11.
Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, Dynamics of network motifs, 2006.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 800

Cf. A000041, A000070, A001477, A026906, A206437.

Rows sums of A133737, A137633, A137679.

a:= n-> add(combinat[numbpart](k), k=1..n): seq(a(n), n=1..44); # Zerinvary Lajos, Jun 01 2008

Table[ Sum[ PartitionsP[k], {k, 1, n}], {n, 1, 45}]
(* or: *)
PartitionsP[Range[45]] // Accumulate (* Jean-François Alcover, Jun 19 2019 *)
CoefficientList[Series[(QPochhammer[x] - 1)/(x (x - 1) QPochhammer[x]), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 29 2022 *)

a(n) = sum(k=1, n, numbpart(k)); \\ Michel Marcus, Jul 19 2023

from sympy import partition
def A026905(n): return sum(partition(k) for k in range(1,n+1)) # Chai Wah Wu, Nov 23 2024

a(n) = A000070(n) - 1, n >= 1.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Oct 25 2016
G.f.: -1/(1 - x) + (1/(1 - x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Dec 25 2016

Edited by N. J. A. Sloane, Jun 20 2015

Name clarified by Omar E. Pol, Apr 30 2022

A036469 Partial sums of A000009 (partitions into distinct parts).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 43, 55, 70, 88, 110, 137, 169, 207, 253, 307, 371, 447, 536, 640, 762, 904, 1069, 1261, 1483, 1739, 2035, 2375, 2765, 3213, 3725, 4310, 4978, 5738, 6602, 7584, 8697, 9957, 11383, 12993, 14809, 16857, 19161, 21751, 24661

Offset: 0

N. J. A. Sloane

nonn

Also number of 1's in all partitions of n+1 into odd parts. Example: a(4)=7 because the partitions of 5 into odd parts are [5], [3,1,1], [1,1,1,1,1], having a total number of 7 1's. - Emeric Deutsch, Mar 29 2006
Convolved with A035363 = A000070. - Gary W. Adamson, Jun 09 2009
Equals row sums of triangle A166240. - Gary W. Adamson, Oct 09 2009
a(n) = if n <= 1 then A201377(1,n) else A201377(n,1). - Reinhard Zumkeller, Dec 02 2011
a(n) equals the sum of the parts of the form 2^k (k >= 0) in all partitions of n + 1 into distinct parts. Example: a(6) = 14. The partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] having sum over parts of the form 2^k equal to 1 + 2 + 4 + 4 + 2 + 1 = 14. - Peter Bala, Dec 01 2013
Number of partitions of the (n+1)-multiset {0,...,0,1} with n 0's into distinct multisets; a(3) = 5: 0|00|1, 00|01, 000|1, 0|001, 0001.  Also number of factorizations of 3*2^n into distinct factors; a(3) = 5: 2*3*4, 4*6, 3*8, 2*12, 24. - Alois P. Heinz, Jul 30 2021

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
A. V. Chekhonadskikh, Some classical number sequences in control system design, Siberian Electronic Mathematical Reports, Volume 14, p. 620-628. See Theorem 2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 774

Cf. A000009, A265093.
Cf. A035363, A000070. - Gary W. Adamson, Jun 09 2009
Cf. A166240. - Gary W. Adamson, Oct 09 2009
Column k=1 of A346520.

g:=1/(1-x)/product(1-x^(2*j-1),j=1..30): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=0..46); # Emeric Deutsch, Mar 29 2006
# second Maple program:
b:= proc(n, i) b(n, i):= `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= proc(n) option remember; b(n, n) +`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

CoefficientList[ Series[Product[(1 + t^i), {i, 1, Infinity}]/(1 - t), {t, 0, 46}], t] (* Geoffrey Critzer, May 16 2010 *)
b[n_, i_] := If[n == 0, 1, If[i<1, 0, b[n, i-1]+If[i>n, 0, b[n-i, Min[n-i, i-1]]]]]; a[n_] := a[n] = b[n, n]+If[n>0, a[n-1], 0]; Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
Accumulate[Table[PartitionsQ[n], {n, 0, 50}]] (* Vaclav Kotesovec, Oct 26 2016 *)

G.f.: 1/[(1-x)*product(1-x^(2j-1), j=1..infinity)]. - Emeric Deutsch, Mar 29 2006
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)) * (1 + (18+13*Pi^2) / (48*Pi*sqrt(3*n)) + (2916 - 1404*Pi^2 + 121*Pi^4)/(13824*Pi^2*n)). - Vaclav Kotesovec, Feb 26 2015, updated Oct 26 2016
For n > 0, a(n) = A026906(n) + 1. - Vaclav Kotesovec, Oct 26 2016
Faster converging g.f.: A(x) = (1/(1 - x))*Sum_{n >= 0} x^(n*(2*n-1))/Product_{k = 1..2*n} (1 - x^k). - Peter Bala, Feb 02 2021

A342086 Number of strict factorizations of divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 5, 3, 5, 2, 9, 2, 5, 5, 7, 2, 9, 2, 9, 5, 5, 2, 16, 3, 5, 5, 9, 2, 15, 2, 10, 5, 5, 5, 18, 2, 5, 5, 16, 2, 15, 2, 9, 9, 5, 2, 25, 3, 9, 5, 9, 2, 16, 5, 16, 5, 5, 2, 31, 2, 5, 9, 14, 5, 15, 2, 9, 5, 15, 2, 34, 2, 5, 9, 9, 5, 15, 2, 25, 7, 5

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

A strict factorization of n is a set of distinct positive integers > 1 with product n.

			The a(1) = 1 through a(12) = 9 factorizations:
  ()  ()   ()   ()   ()   ()     ()   ()     ()   ()     ()    ()
      (2)  (3)  (2)  (5)  (2)    (7)  (2)    (3)  (2)    (11)  (2)
                (4)       (3)         (4)    (9)  (5)          (3)
                          (6)         (8)         (10)         (4)
                          (2*3)       (2*4)       (2*5)        (6)
                                                               (12)
                                                               (2*3)
                                                               (2*6)
                                                               (3*4)

Robert Israel, Table of n, a(n) for n = 1..10000

A version for partitions is A026906 (strict partitions of 1..n).
A version for partitions is A036469 (strict partitions of 0..n).
A version for partitions is A047966 (strict partitions of divisors).
The non-strict version is A057567.
A000005 counts divisors, with sum A000203.
A000009 counts strict partitions.
A001055 counts factorizations, with strict case A045778.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
Cf. A001227, A050320, A340101, A340596, A340654, A340655, A340853, A341596, A341673, A341674, A342097.

sf1:= proc(n,m)
  local D,d;
  if n = 1 then return 1 fi;
  D:= select(`<`,numtheory:-divisors(n) minus {1},m);
  add( procname(n/d,d), d= D)
end proc:
sf:= proc(n) option remember; sf1(n,n+1) end proc:f:= proc(n) local d; add(sf(d),d=numtheory:-divisors(n)) end proc:map(f, [$1..100]); # Robert Israel, Mar 10 2021

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Select[facs[k],UnsameQ@@#&]],{k,Divisors[n]}],{n,30}]

A277643 Partial sums of number of overpartitions (A015128).

Original entry on oeis.org

1, 3, 7, 15, 29, 53, 93, 157, 257, 411, 643, 987, 1491, 2219, 3259, 4731, 6793, 9657, 13605, 19005, 26341, 36245, 49533, 67261, 90789, 121855, 162679, 216087, 285655, 375903, 492527, 642671, 835283, 1081539, 1395347, 1793987, 2298873, 2936465, 3739401, 4747849

Offset: 0

Vaclav Kotesovec, Oct 25 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A000070, A036469, A026906.

Cf. A211971, A002865, A087897.

Accumulate[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]]
nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)

a(n) = Sum_{k=0..n} A015128(k).
a(n) ~ exp(Pi*sqrt(n))/(4*Pi*sqrt(n)) * (1 + Pi/(4*sqrt(n))).
G.f.: 1/(1-x) * Product_{k>=1} (1 + x^k) / (1 - x^k). - Vaclav Kotesovec, Mar 25 2017
G.f.: 1/((1 - x)*theta_4(x)), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

A317910 Expansion of -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} (1 + x^k).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 11, 16, 23, 32, 43, 57, 74, 95, 121, 152, 189, 234, 287, 350, 425, 513, 616, 737, 878, 1042, 1233, 1454, 1709, 2004, 2343, 2732, 3179, 3690, 4274, 4941, 5700, 6563, 7544, 8656, 9915, 11340, 12949, 14764, 16811, 19114, 21703, 24612, 27875, 31532, 35628, 40209

Offset: 0

Ilya Gutkovskiy, Aug 10 2018

nonn

Partial sums of A111133.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
Riccardo Aragona, Roberto Civino, Norberto Gavioli, Carlo Maria Scoppola, A Chain of Normalizers in the Sylow _2-subgroups of the symmetric group on 2^n letters, arXiv:2008.13423 [math.GR], 2020.
Riccardo Aragona, Roberto Civino, Norberto Gavioli, and Carlo Maria Scoppola, Rigid commutators and a normalizer chain, arXiv:2009.11149 [math.GR], 2020.
Index entries for sequences related to partitions

Cf. A026906, A036469, A058682, A111133.

a:=series(-1/(1-x)^2+(1/(1-x))*mul((1 + x^k),k=1..100),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

nmax = 52; CoefficientList[Series[-1/(1 - x)^2 + 1/(1 - x) Product[1 + x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* or *)
nmax = 52; CoefficientList[Series[1/((1 - x) QPochhammer[x, x^2]) - 1/(1 - x)^2, {x, 0, nmax}], x] (* or *)
Table[Sum[PartitionsQ[k] - 1, {k, 0, n}] , {n, 0, 52}]

G.f.: -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} 1/(1 - x^(2*k-1)).
a(n) = A036469(n) - n - 1.
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (2*Pi*n^(1/4)). - Vaclav Kotesovec, Aug 21 2018

A262885 Irregular triangle T(n,k) read by rows: T(n,k) = number of partitions of n into at least two distinct parts, where the largest part is n-k.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 1, 2, 2, 3, 3, 2, 1, 1, 2, 2, 3, 4, 3, 1, 1, 1, 2, 2, 3, 4, 4, 3, 1, 1, 1, 2, 2, 3, 4, 5, 4, 3, 1, 1, 1, 2, 2, 3, 4, 5, 5, 5, 3, 1, 1, 2, 2, 3, 4, 5, 6, 6, 5, 2, 1, 1, 2, 2, 3, 4, 5, 6, 7, 7, 5, 2

Offset: 1

Bob Selcoe, Oct 04 2015

Alternate name: T(n,k) = the number of ways that at least two distinct positive integers sum to n, where the largest of these integers is n-k.
Row sums = A111133(n).
Row sums {k <= floor((n-1)/2)} = A026906(n)
Row sums {k > floor((n-1)/2)} = A258259(n)

			Triangle starts T(1,1):
n/k  1 2 3 4 5 6 7 8 9 10 11 12 13 14
1    0
2    0
3    1
4    1
5    1 1
6    1 1 1
7    1 1 2
8    1 1 2 1
9    1 1 2 2 1
10   1 1 2 2 2 1
11   1 1 2 2 3 2
12   1 1 2 2 3 3 2
13   1 1 2 2 3 4 3 1
14   1 1 2 2 3 4 4 3 1
15   1 1 2 2 3 4 5 4 3 1
16   1 1 2 2 3 4 5 5 5 3
17   1 1 2 2 3 4 5 6 6 5  2
18   1 1 2 2 3 4 5 6 7 7  5  2
19   1 1 2 2 3 4 5 6 8 8  7  5  1
20   1 1 2 2 3 4 5 6 8 9  9  8  4  1
T(15,8) = 4: the four partitions of 15 into at least two distinct parts with largest part 15-8 = 7 are  {7,6,2}; {7,5,3}; {7,5,2,1} and {7,4,3,1}.
T(14,k) for k=1..F, with F = floor(13/2) = 6: T(14,1) = 0+1 = 1; T(14,2) = 0+1 = 1; T(14,3) = 1+1 = 2; T(14,4) = 1+1 = 2; T(14,5) = 2+1 = 3; T(14,6) = 3+1 = 4.
T(14,k) for k>6: T(14,7) = T(7,1)+T(7,2)+T(7,3) = 1+1+2 = 4; T(14,8) = T(8,3)+T(8,4) = 2+1 = 3; T(14,9) = T(9,5) = 1.

Cf. A000009, A026906, A111133, A258259.

Given T(1,1) = T(2,1) = 0, to find row n>=3: Let k" be the maximum value of k in row g

T(n,k) = S(g)+1 g=k when g<=F (equivalent to A000009(g));

T(n,k) = Sum_{j=2*(g-F)-1..k"} T(g,j) g=k when g>F, 2*(g-F)-1 <= k" and n is even;

T(n,k) = Sum_{j=2*(g-F)..k"} T(g,j) g=k when g>F, 2*(g-F) <= k" and n is odd.

Showing 1-9 of 9 results.

cp's OEIS Frontend

A056870 Difference between partial sums of partition numbers (A026905) and partial sums of numbers of partitions into distinct parts (A026906).

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A056871 Sum of partial sums of partition numbers (A026905) and partial sums of numbers of partitions into distinct parts (A026906).

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

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A026905 Partial sums of the partition numbers A000041 of the positive integers.

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A036469 Partial sums of A000009 (partitions into distinct parts).

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A342086 Number of strict factorizations of divisors of n.

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A277643 Partial sums of number of overpartitions (A015128).

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A317910 Expansion of -1/(1 - x)^2 + (1/(1 - x))*Product_{k>=1} (1 + x^k).

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