A025147 -id:A025147 - cp's OEIS frontend

A002865 Number of partitions of n that do not contain 1 as a part.

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701

Offset: 0

N. J. A. Sloane

Also the number of partitions of n-1, n >= 2, such that the least part occurs exactly once. See A096373, A097091, A097092, A097093. - Robert G. Wilson v, Jul 24 2004 [Corrected by Wolfdieter Lang, Feb 18 2009]
Number of partitions of n+1 where the number of parts is itself a part. Take a partition of n (with k parts) which does not contain 1, remove 1 from each part and add a new part of size k+1. - Franklin T. Adams-Watters, May 01 2006
Number of partitions where the largest part occurs at least twice. - Joerg Arndt, Apr 17 2011
Row sums of triangle A147768. - Gary W. Adamson, Nov 11 2008
From Lewis Mammel (l_mammel(AT)att.net), Oct 06 2009: (Start)
a(n) is the number of sets of n disjoint pairs of 2n things, called a pairing, disjoint with a given pairing (A053871), that are unique under permutations preserving the given pairing.
Can be seen immediately from a graphical representation which must decompose into even numbered cycles of 4 or more things, as connected by pairs alternating between the pairings. Each thing is in a single cycle, so this is a partition of 2n into even parts greater than 2, equivalent to a partition of n into parts greater than 1. (End)
Convolution product (1, 1, 2, 2, 4, 4, ...) * (1, 2, 3, ...) = A058682 starting (1, 3, 7, 13, 23, 37, ...); with row sums of triangle A171239 = A058682. - Gary W. Adamson, Dec 05 2009
Also the number of 2-regular multigraphs with loops forbidden. - Jason Kimberley, Jan 05 2011
Number of appearances of the multiplicity n, n-1, ..., n-k in all partitions of n, for k < n/2. (Only populated by multiplicities of large numbers of 1's.) - William Keith, Nov 20 2011
Also the number of equivalence classes of n X n binary matrices with exactly 2 1's in each row and column, up to permutations of rows and columns (cf. A133687). - N. J. A. Sloane, Sep 16 2013
The q-Catalan numbers ((1-q)/(1-q^(n+1)))[2n,n]q, where [2n,n]_q are the central q-binomial coefficients, match this sequence in their initial segment of length n. - _William J. Keith, Nov 14 2013
Starting at a(2) this sequence gives the number of vertices on a nim tree created in the game of edge removal for a path P_{n} where n is the number of vertices on the path. This is the number of nonisomorphic graphs that can result from the path when the game of edge removal is played. - Lyndsey Wong, Jul 09 2016
The number of different ways to climb a staircase taking at least two stairs at a time. - Mohammad K. Azarian, Nov 20 2016
Let 1,0,1,1,1,... (offset 0) count unlabeled, connected, loopless 1-regular digraphs. This here is the Euler transform of that sequence, counting unlabeled loopless 1-regular digraphs. A145574 is the associated multiset transformation. A000166 are the labeled loopless 1-regular digraphs. - R. J. Mathar, Mar 25 2019
For n > 1, also the number of partitions with no part greater than the number of ones. - George Beck, May 09 2019 [See A187219 which is the correct sequence for this interpretation for n >= 1. - Spencer Miller, Jan 30 2023]
From Gus Wiseman, May 19 2019: (Start)
Conjecture: Also the number of integer partitions of n - 1 that have a consecutive subsequence summing to each positive integer from 1 to n - 1. For example, (32211) is such a partition because we have consecutive subsequences:
  1: (1)
  2: (2)
  3: (3) or (21)
  4: (22) or (211)
  5: (32) or (221)
  6: (2211)
  7: (322)
  8: (3221)
  9: (32211)
(End)
There is a sufficient and necessary condition to characterize the partitions defined by Gus Wiseman. It is that the largest part must be less than or equal to the number of ones plus one. Hence, the number of partitions of n with no part greater than the number of ones is the same as the number of partitions of n-1 that have a consecutive subsequence summing to each integer from 1 to n-1. Gus Wiseman's conjecture can be proved bijectively. - Andrew Yezhou Wang, Dec 14 2019
From Peter Bala, Dec 01 2024: (Start)
Let P(2, n) denote the set of partitions of n into parts k > 1. Then A000041(n) = - Sum_{parts k in all partitions in P(2, n+2)} mu(k). For example, with n = 5, there are 4 partitions of n + 2 = 7 into parts greater than 1, namely, 7, 5 + 2, 4 + 3, 3 + 2 + 2, and mu(7) + (mu(5) + mu(2)) + (mu(4 ) + mu(3)) + (mu(3) + mu(2) + mu(2)) = -7 = - A000041(5). (End)

			a(6) = 4 from 6 = 4+2 = 3+3 = 2+2+2.
G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 7*x^8 + 8*x^9 + ...
From _Gus Wiseman_, May 19 2019: (Start)
The a(2) = 1 through a(9) = 8 partitions not containing 1 are the following. The Heinz numbers of these partitions are given by A005408.
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (22)  (32)  (33)   (43)   (44)    (54)
                        (42)   (52)   (53)    (63)
                        (222)  (322)  (62)    (72)
                                      (332)   (333)
                                      (422)   (432)
                                      (2222)  (522)
                                              (3222)
The a(2) = 1 through a(9) = 8 partitions of n - 1 whose least part appears exactly once are the following. The Heinz numbers of these partitions are given by A247180.
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (31)  (32)   (42)   (43)    (53)
                        (41)   (51)   (52)    (62)
                        (221)  (321)  (61)    (71)
                                      (331)   (332)
                                      (421)   (431)
                                      (2221)  (521)
                                              (3221)
The a(2) = 1 through a(9) = 8 partitions of n + 1 where the number of parts is itself a part are the following. The Heinz numbers of these partitions are given by A325761.
  (21)  (22)  (32)   (42)   (52)    (62)    (72)     (82)
              (311)  (321)  (322)   (332)   (333)    (433)
                            (331)   (431)   (432)    (532)
                            (4111)  (4211)  (531)    (631)
                                            (4221)   (4222)
                                            (4311)   (4321)
                                            (51111)  (4411)
                                                     (52111)
The a(2) = 1 through a(8) = 7 partitions of n whose greatest part appears at least twice are the following. The Heinz numbers of these partitions are given by A070003.
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (11111)  (222)     (2221)     (332)
                                (2211)    (22111)    (2222)
                                (111111)  (1111111)  (3311)
                                                     (22211)
                                                     (221111)
                                                     (11111111)
Nonisomorphic representatives of the a(2) = 1 through a(6) = 4 2-regular multigraphs with n edges and n vertices are the following.
  {12,12}  {12,13,23}  {12,12,34,34}  {12,12,34,35,45}  {12,12,34,34,56,56}
                       {12,13,24,34}  {12,13,24,35,45}  {12,12,34,35,46,56}
                                                        {12,13,23,45,46,56}
                                                        {12,13,24,35,46,56}
The a(2) = 1 through a(9) = 8 partitions of n with no part greater than the number of ones are the following. The Heinz numbers of these partitions are given by A325762.
  (11)  (111)  (211)   (2111)   (2211)    (22111)    (22211)     (33111)
               (1111)  (11111)  (3111)    (31111)    (32111)     (222111)
                                (21111)   (211111)   (41111)     (321111)
                                (111111)  (1111111)  (221111)    (411111)
                                                     (311111)    (2211111)
                                                     (2111111)   (3111111)
                                                     (11111111)  (21111111)
                                                                 (111111111)
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 836.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, p*(n).
H. P. Robinson, Letter to N. J. A. Sloane, Jan 04 1974.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 334.

Andrew van den Hoeven, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. P. Akande et al., Computational study of non-unitary partitions, arXiv:2112.03264 [math.CO], 2021.
Colin Albert, Olivia Beckwith, Irfan Demetoglu, Robert Dicks, John H. Smith, and Jasmine Wang, Integer partitions with large Dyson rank, arXiv:2203.08987 [math.NT], 2022.
Max A. Alekseyev and Allan Bickle, Forbidden Subgraphs of Single Graphs, (2024). See p. 6.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
G. Dahl and T. A. Haufmann, Zero-one completely positive matrices and the A(R,S) classes, Preprint, 2016.
Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.
R. P. Gallant, G. Gunther, B. L. Hartnell, and D. F. Rall, A game of edge removal on graphs, JCMCC, 57 (2006), 75 - 82.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
H. Gropp, On tactical configurations, regular bipartite graphs and (v,k,even)-designs, Discr. Math., 155 (1996), 81-98.
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 100
Wenwei Li, Approximation of the Partition Number After Hardy and Ramanujan: An Application of Data Fitting Method in Combinatorics, arXiv preprint arXiv:1612.05526 [math.NT], 2016-2018.
Wenwei Li, On the Number of Conjugate Classes of Derangements, arXiv:1612.08186 [math.CO], 2016.
J. L. Nicolas and A. Sárközy, On partitions without small parts, Journal de théorie des nombres de Bordeaux, 12 no. 1 (2000), p. 227-254.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
H. P. Robinson, Letter to N. J. A. Sloane, Jul 12 1971
H. P. Robinson, Letter to N. J. A. Sloane, Dec 10 1973
H. P. Robinson, Letter to N. J. A. Sloane, Jan 4 1974.
Noah Rubin, Curtis Bright, Kevin K. H. Cheung, and Brett Stevens, Integer and Constraint Programming Revisited for Mutually Orthogonal Latin Squares, arXiv:2103.11018 [cs.DM], 2021.
Miloslav Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.
Miloslav Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.
Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.
Index entries for related partition-counting sequences

First differences of partition numbers A000041. Cf. A053445, A072380, A081094, A081095, A232697.
Pairwise sums seem to be in A027336.
Essentially the same as A085811.
Cf. A025147, A147768, A058682, A171239.
A column of A090824 and of A133687 and of A292508 and of A292622. Cf. A229161.
2-regular not necessarily connected graphs: A008483 (simple graphs), A000041 (multigraphs with loops allowed), this sequence (multigraphs with loops forbidden), A027336 (graphs with loops allowed but no multiple edges). - Jason Kimberley, Jan 05 2011
See also A098743 (parts that do not divide n).
Numbers n such that in the edge-delete game on the path P_{n} the first player does not have a winning strategy: A274161. - Lyndsey Wong, Jul 09 2016
Cf. A002033, A070003, A103295, A247180, A325676, A325684, A325761, A325762, A325763.
Row sums of characteristic array A145573.
Number of partitions of n into parts >= m: A008483 (m = 3), A008484 (m = 4), A185325 - A185329 (m = 5 through 9).

Concatenation([1],List([1..41],n->NrPartitions(n)-NrPartitions(n-1))); # Muniru A Asiru, Aug 20 2018

A41 := func; [A41(n)-A41(n-1):n in [0..50]]; // Jason Kimberley, Jan 05 2011

with(combstruct): ZL1:=[S, {S=Set(Cycle(Z,card>1))}, unlabeled]: seq(count(ZL1,size=n), n=0..50);  # Zerinvary Lajos, Sep 24 2007
G:= {P=Set (Set (Atom, card>1))}: combstruct[gfsolve](G, unlabeled, x): seq  (combstruct[count] ([P, G, unlabeled], size=i), i=0..50);  # Zerinvary Lajos, Dec 16 2007
with(combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, unlabeled]; end: A:=a(2):seq(count(A, size=n), n=0..50);  # Zerinvary Lajos, Jun 11 2008
# alternative Maple program:
A002865:= proc(n) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)-1)*A002865(n-j), j=1..n)/n)
    end:
seq(A002865(n), n=0..60);  # Alois P. Heinz, Sep 17 2017

Table[ PartitionsP[n + 1] - PartitionsP[n], {n, -1, 50}] (* Robert G. Wilson v, Jul 24 2004 *)
f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k == n, 1, f[n, k + 1] + f[n - k, k]]]]; Table[ f[n, 2], {n, 50}] (* Robert G. Wilson v *)
Table[SeriesCoefficient[Exp[Sum[x^(2*k)/(k*(1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Aug 18 2018 *)
CoefficientList[Series[1/QPochhammer[x^2, x], {x,0,50}], x] (* G. C. Greubel, Nov 03 2019 *)
Table[Count[IntegerPartitions[n],?(FreeQ[#,1]&)],{n,0,50}] (* _Harvey P. Dale, Feb 12 2023 *)

{a(n) = if( n<0, 0, polcoeff( (1 - x) / eta(x + x * O(x^n)), n))};

a(n)=if(n,numbpart(n)-numbpart(n-1),1) \\ Charles R Greathouse IV, Nov 26 2012

from sympy import npartitions
def A002865(n): return npartitions(n)-npartitions(n-1) if n else 1 # Chai Wah Wu, Mar 30 2023

def A002865_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/product((1-x^(m+2)) for m in (0..60)) ).list()
A002865_list(50) # G. C. Greubel, Nov 03 2019

G.f.: Product_{m>1} 1/(1-x^m).
a(0)=1, a(n) = p(n) - p(n-1), n >= 1, with the partition numbers p(n) := A000041(n).
a(n) = A085811(n+3). - James Sellers, Dec 06 2005 [Corrected by Gionata Neri, Jun 14 2015]
a(n) = A116449(n) + A116450(n). - Reinhard Zumkeller, Feb 16 2006
a(n) = Sum_{k=2..floor((n+2)/2)} A008284(n-k+1,k-1) for n > 0. - Reinhard Zumkeller, Nov 04 2007
G.f.: 1 + Sum_{n>=2} x^n / Product_{k>=n} (1 - x^k). - Joerg Arndt, Apr 13 2011
G.f.: Sum_{n>=0} x^(2*n) / Product_{k=1..n} (1 - x^k). - Joerg Arndt, Apr 17 2011
a(n) = A090824(n,1) for n > 0. - Reinhard Zumkeller, Oct 10 2012
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (217*Pi^2/6912 + 9/(2*Pi^2) + 13/8)/n). - Vaclav Kotesovec, Feb 26 2015, extended Nov 04 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(0) = 1, a(n) = A232697(n) - 1. - George Beck, May 09 2019
From Peter Bala, Feb 19 2021: (Start)
G.f.: A(q) = Sum_{n >= 0} q^(n^2)/( (1 - q)*Product_{k = 2..n} (1 - q^k)^2 ).
More generally, A(q) = Sum_{n >= 0} q^(n*(n+r))/( (1 - q) * Product_{k = 2..n} (1 - q^k)^2 * Product_{i = 1..r} (1 - q^(n+i)) ) for r = 0,1,2,.... (End)
G.f.: 1 + Sum_{n >= 1} x^(n+1)/Product_{k = 1..n-1} 1 - x^(k+2). - Peter Bala, Dec 01 2024

A087897 Number of partitions of n into odd parts greater than 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 8, 8, 10, 12, 13, 15, 18, 20, 23, 27, 30, 34, 40, 44, 50, 58, 64, 73, 83, 92, 104, 118, 131, 147, 166, 184, 206, 232, 256, 286, 320, 354, 394, 439, 485, 538, 598, 660, 730, 809, 891, 984, 1088, 1196, 1318, 1454, 1596, 1756

Offset: 0

N. J. A. Sloane, Dec 04 2003

Also number of partitions of n into distinct parts which are not powers of 2.
Also number of partitions of n into distinct parts such that the two largest parts differ by 1.
Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006
Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007
In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013
Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014
From Gus Wiseman, Aug 22 2021: (Start)
Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:
  32  222  322  332   432   3322   3332   4332    4432    5432     43332
                2222  3222  22222  4322   33222   33322   33332    44322
                                   32222  222222  43222   43322    333222
                                                  322222  332222   432222
                                                          2222222  3222222
(End)

			1 + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 2*x^11 + 3*x^12 + 3*x^13 + ...
q + q^73 + q^121 + q^145 + q^169 + q^193 + 2*q^217 + 2*q^241 + 2*q^265 + ...
a(10)=2 because we have [7,3] and [5,5].
From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(22)=13 symmetric unimodal compositions of 22+3=25 where the maximal part appears three times:
01:  [ 1 1 1 1 1 1 1 1 3 3 3 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 2 3 3 3 2 1 1 1 1 1 1 ]
03:  [ 1 1 1 1 1 5 5 5 1 1 1 1 1 ]
04:  [ 1 1 1 1 2 2 3 3 3 2 2 1 1 1 1 ]
05:  [ 1 1 1 2 5 5 5 2 1 1 1 ]
06:  [ 1 1 2 2 2 3 3 3 2 2 2 1 1 ]
07:  [ 1 1 3 5 5 5 3 1 1 ]
08:  [ 1 1 7 7 7 1 1 ]
09:  [ 1 2 2 5 5 5 2 2 1 ]
10:  [ 1 4 5 5 5 4 1 ]
11:  [ 2 2 2 2 3 3 3 2 2 2 2 ]
12:  [ 2 3 5 5 5 3 2 ]
13:  [ 2 7 7 7 2 ]
(End)
From _Gus Wiseman_, Feb 16 2021: (Start)
The a(7) = 1 through a(19) = 8 partitions are the following (A..J = 10..19). The Heinz numbers of these partitions are given by A341449.
  7  53  9    55  B    75    D    77    F      97    H      99      J
         333  73  533  93    553  95    555    B5    755    B7      775
                       3333  733  B3    753    D3    773    D5      955
                                  5333  933    5533  953    F3      973
                                        33333  7333  B33    5553    B53
                                                     53333  7533    D33
                                                            9333    55333
                                                            333333  73333
(End)

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. I

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from Alois P. Heinz)
C. Ballantine and M. Merca, Padovan numbers as sums over partitions into odd parts, Journal of Inequalities and Applications, (2016) 2016:1; doi.
B. C. Berndt, B. Kim, and A. J. Yee, Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions, J. Comb. Thy. Ser. A, 117 (2010), 957-973.
Howard D. Grossman, Problem 228, Mathematics Magazine, 28 (1955), p. 160.
R. K. Guy, Two theorems on partitions, Math. Gaz., 42 (1958), 84-86. Math. Rev. 20 #3110.
Cristiano Husu, The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2, arXiv:1804.09883 [math.NT], 2018.
James Mc Laughlin, Andrew V. Sills, and Peter Zimmer, Rogers-Ramanujan-Slater Type Identities, Electronic J. Combinatorics, DS15, 1-59, May 31, 2008.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

The ordered version is A000931.
Partitions with no ones are counted by A002865, ranked by A005408.
The even version is A035363, ranked by A066207.
The version for factorizations is A340101.
Partitions whose only even part is the smallest are counted by A341447.
The Heinz numbers of these partitions are given by A341449.
A000009 counts partitions into odd parts, ranked by A066208.
A025147 counts strict partitions with no 1's.
A025148 counts strict partitions with no 1's or 2's.
A026804 counts partitions whose smallest part is odd, ranked by A340932.
A027187 counts partitions with even length/maximum, ranks A028260/A244990.
A027193 counts partitions with odd length/maximum, ranks A026424/A244991.
A058695 counts partitions of odd numbers, ranked by A300063.
A058696 counts partitions of even numbers, ranked by A300061.
A340385 counts partitions with odd length and maximum, ranked by A340386.
Cf. A000041, A003114, A039900, A160786, A257991/A257992, A264396, A300272, A339662, A339737, A339886.

a087897 = p [3,5..] where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

To get 128 terms: t4 := mul((1+x^(2^n)),n=0..7); t5 := mul((1+x^k),k=1..128): t6 := series(t5/t4,x,100); t7 := seriestolist(t6);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<3, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 11 2013

max = 65; f[x_] := Product[ 1/(1 - x^(2k+1)), {k, 1, max}]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Dec 16 2011, after Emeric Deutsch *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<3, 0, b[n, i-2]+If[i>n, 0, b[n-i, i]]] ]; a[n_] := b[n, n-1+Mod[n, 2]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Apr 01 2015, after Alois P. Heinz *)
Flatten[{1, Table[PartitionsQ[n+1] - PartitionsQ[n], {n, 0, 80}]}] (* Vaclav Kotesovec, Dec 01 2015 *)
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&OddQ[Times@@#]&]],{n,0,30}] (* Gus Wiseman, Feb 16 2021 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - x) * eta(x^2 + A) / eta(x + A), n))} /* Michael Somos, Nov 13 2011 */

from functools import lru_cache
@lru_cache(maxsize=None)
def A087897_T(n,k):
    if n==0: return 1
    if k<3 or n<0: return 0
    return A087897_T(n,k-2)+A087897_T(n-k,k)
def A087897(n): return A087897_T(n,n-(n&1^1)) # Chai Wah Wu, Sep 23 2023, after Alois P. Heinz

Expansion of q^(-1/24) * (1 - q) * eta(q^2) / eta(q) in powers of q.
Expansion of (1 - x) / chi(-x) in powers of x where chi() is a Ramanujan theta function.
G.f.: 1 + x^3 + x^5*(1 + x) + x^7*(1 + x)*(1 + x^2) + x^9*(1 + x)*(1 + x^2)*(1 + x^3) + ... [Glaisher 1876]. - Michael Somos, Jun 20 2012
G.f.: Product_{k >= 1} 1/(1-x^(2*k+1)).
G.f.: Product_{k >= 1, k not a power of 2} (1+x^k).
G.f.: Sum_{k >= 1} x^(3*k)/Product_{j = 1..k} (1 - x^(2*j)). - Emeric Deutsch, Mar 30 2006
a(n) ~ exp(Pi*sqrt(n/3)) * Pi / (8 * 3^(3/4) * n^(5/4)) * (1 - (15*sqrt(3)/(8*Pi) + 11*Pi/(48*sqrt(3)))/sqrt(n) + (169*Pi^2/13824 + 385/384 + 315/(128*Pi^2))/n). - Vaclav Kotesovec, Aug 30 2015, extended Nov 04 2016
G.f.: 1/(1 - x^3) * Sum_{n >= 0} x^(5*n)/Product_{k = 1..n} (1 - x^(2*k)) = 1/((1 - x^3)*(1 - x^5)) * Sum_{n >= 0} x^(7*n)/Product_{k = 1..n} (1 - x^(2*k)) = ..., extending Deutsch's result dated Mar 30 2006. - Peter Bala, Jan 15 2021
G.f.: Sum_{n >= 0} x^(n*(2*n+1))/Product_{k = 2..2*n+1} (1 - x^k). (Set z = x^3 and q = x^2 in Mc Laughlin et al., Section 1.3, Entry 7.) - Peter Bala, Feb 02 2021
a(2*n+1) = Sum{j>=1} A008284(n+1-j,2*j - 1) and a(2*n) = Sum{j>=1} A008284(n-j, 2*j). - Gregory L. Simay, Sep 22 2023

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Gus Wiseman, Sep 17 2023

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)

Robert Price, Table of n, a(n) for n = 2..1226
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

A318029 Expansion of Sum_{k>=2} x^(k*(k+3)/2) / Product_{j=1..k} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 4, 6, 7, 9, 11, 14, 16, 20, 24, 28, 34, 40, 47, 55, 65, 75, 88, 102, 118, 136, 158, 180, 208, 238, 272, 311, 355, 403, 459, 521, 590, 668, 756, 852, 962, 1084, 1218, 1370, 1538, 1724, 1932, 2163, 2417, 2701, 3015, 3361, 3745, 4170, 4636, 5154, 5724

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Number of partitions of n into at least two distinct parts >= 2.

			a(9) = 4 because we have [7, 2], [6, 3], [5, 4] and [4, 3, 2].

Index entries for sequences related to partitions

Cf. A000009, A025147, A083751, A096765, A111133.

nmax = 60; CoefficientList[Series[Sum[x^(k (k + 3)/2)/Product[(1 - x^j), {j, 1, k}], {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[x - 1/(1 - x) + 1/((1 + x) QPochhammer[x, x^2]), {x, 0, nmax}], x]
Join[{0, 0}, Table[-1 + Sum[(-1)^(n - k) PartitionsQ[k], {k, 0, n}], {n, 2, 60}]]

G.f.: x - 1/(1 - x) + Product_{k>=2} (1 + x^k).

a(n) = A025147(n) - 1 for n > 1.

A379722 Numbers whose prime indices do not have the same sum as product.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Gus Wiseman, Jan 08 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Partitions of this type are counted by A379736.
The complement is A301987, counted by A001055.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

Nonzeros of A325036.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
A379681 gives sum plus product of prime indices, firsts A379682.
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721
- different: A379736, ranks A379722 (this)
Cf. A000720, A003963, A025147, A075254, A111133, A178503, A319000, A325041.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@prix[#]!=Total[prix[#]]&]

A307719 Number of partitions of n into 3 mutually coprime parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 2, 7, 2, 8, 4, 8, 4, 15, 4, 16, 7, 15, 7, 26, 7, 23, 11, 26, 10, 43, 9, 35, 16, 38, 16, 54, 14, 49, 23, 54, 18, 79, 18, 66, 31, 64, 25, 100, 25, 89, 36, 85, 31, 127, 35, 104, 46, 104, 39, 167, 36, 125, 58, 129, 52, 185, 45

Offset: 0

Wesley Ivan Hurt, Apr 24 2019

The Heinz numbers of these partitions are the intersection of A014612 (triples) and A302696 (pairwise coprime). - Gus Wiseman, Oct 16 2020

			There are 2 partitions of 9 into 3 mutually coprime parts: 7+1+1 = 5+3+1, so a(9) = 2.
There are 4 partitions of 10 into 3 mutually coprime parts: 8+1+1 = 7+2+1 = 5+4+1 = 5+3+2, so a(10) = 4.
There are 2 partitions of 11 into 3 mutually coprime parts: 9+1+1 = 7+3+1, so a(11) = 2.
There are 7 partitions of 12 into 3 mutually coprime parts: 10+1+1 = 9+2+1 = 8+3+1 = 7+4+1 = 6+5+1 = 7+3+2 = 5+4+3, so a(12) = 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)
Index entries for sequences related to partitions

A023022 is the version for pairs.
A220377 is the strict case, with ordered version A220377*6.
A327516 counts these partitions of any length, with strict version A305713 and Heinz numbers A302696.
A337461 is the ordered version.
A337563 is the case with no 1's.
A337599 is the pairwise non-coprime instead of pairwise coprime version.
A337601 only requires the distinct parts to be pairwise coprime.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A007304, A014612, A078374, A082024, A284825, A304709, A337462, A337605.

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
for a from 1 to N/3 do
  for b from a to (N-a)/2 do
    if igcd(a,b) > 1 then next fi;
    ab:= a*b;
    for c from b to N-a-b do
       if igcd(ab,c)=1 then A[a+b+c]:= A[a+b+c]+1 fi
od od od:
convert(A,list); # Robert Israel, May 09 2019

Table[Sum[Sum[Floor[1/(GCD[i, j] GCD[j, n - i - j] GCD[i, n - i - j])], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,100}] (* Gus Wiseman, Oct 15 2020 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [gcd(i,j) * gcd(j,n-i-j) * gcd(i,n-i-j) = 1], where [] is the Iverson bracket.

a(n > 2) = A220377(n) + 1. - Gus Wiseman, Oct 15 2020

A379736 Number of integer partitions of n whose product of parts is not n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 9, 14, 19, 28, 40, 55, 73, 100, 133, 174, 226, 296, 381, 489, 623, 790, 1000, 1254, 1568, 1956, 2434, 3007, 3714, 4564, 5599, 6841, 8342, 10141, 12308, 14881, 17968, 21636, 26013, 31183, 37331, 44582, 53169, 63260, 75171, 89130, 105556

Offset: 0

Gus Wiseman, Jan 07 2025

nonn

These partitions are ranked by A379722, complement A301987.

			The a(2) = 1 through a(7) = 14 partitions:
  (11)  (21)   (31)    (32)     (33)      (43)
        (111)  (211)   (41)     (42)      (52)
               (1111)  (221)    (51)      (61)
                       (311)    (222)     (322)
                       (2111)   (411)     (331)
                       (11111)  (2211)    (421)
                                (3111)    (511)
                                (21111)   (2221)
                                (111111)  (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

The complement is counted by A001055.
The strict case is A111133 (except first term).
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1, see A379734, strict A379735.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736 (this), ranks A379722
Cf. A028422, A069016, A319000, A319916, A325036, A325041, A326152, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],Times@@#!=n&]],{n,0,30}]

a(n) = A000041(n) - A001055(n).

A219282 Number of superdiagonal bargraphs with area n.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 49, 68, 93, 126, 170, 229, 308, 413, 551, 731, 965, 1269, 1664, 2177, 2842, 3701, 4806, 6222, 8031, 10337, 13272, 17003, 21740, 27745, 35343, 44936, 57021, 72213, 91274, 115149, 145010, 182309, 228841, 286819, 358964, 448614, 559857, 697694

Offset: 0

Joerg Arndt, Dec 04 2012

nonn

Number of compositions n = p(1) + p(2) + ... + p(m) such that p(k) >= k (superdiagonal compositions), see example. - Joerg Arndt, Dec 19 2012

Number of (n-2)-bit binary strings in which the runs of ones are successively (1, 11, 111, 1111, ...), as in for example 00101100111011110011111000... To turn such a string into a composition, add 'X0 to the start of the empty string and the mark ' to the end, replace the runs 1, 11, 111,... with '01, '011, '0111, ... then consider the distances between the marks. - Andrew Woods, Jan 02 2015

			From _Joerg Arndt_, Dec 19 2012: (Start)
The a(9) = 18 compositions 9 = p(1) + p(2) + ... + p(m) such that p(k) >= k are
[ 1]  [ 1 2 6 ]
[ 2]  [ 1 3 5 ]
[ 3]  [ 1 4 4 ]
[ 4]  [ 1 5 3 ]
[ 5]  [ 1 8 ]
[ 6]  [ 2 2 5 ]
[ 7]  [ 2 3 4 ]
[ 8]  [ 2 4 3 ]
[ 9]  [ 2 7 ]
[10]  [ 3 2 4 ]
[11]  [ 3 3 3 ]
[12]  [ 3 6 ]
[13]  [ 4 2 3 ]
[14]  [ 4 5 ]
[15]  [ 5 4 ]
[16]  [ 6 3 ]
[17]  [ 7 2 ]
[18]  [ 9 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Margaret Archibald, Aubrey Blecher, Arnold Knopfmacher, and Stephan Wagner, Subdiagonal and superdiagonal compositions, Art Disc. Appl. Math. (2024). See p. 10.
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Cf. A063978 (compositions such that p(k) >= k-1 for k >= 2).
Cf. A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A098131 (compositions with smallest part >= number of parts; g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).
Cf. A143862 (compositions with every part divisible by number of parts; g.f. Sum_{k>=0} x^(k^2) / (1 - x^k)^k).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Row sums of A305556.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, i+1), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 28 2014

nmax = 50; CoefficientList[Series[Sum[x^(k*(k+1)/2) / (1-x)^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 05 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

N=66; q='q+O('q^N);
gf=sum(n=0,N, q^(n*(n+1)/2) / (1-q)^n );
v=Vec(gf)

G.f.: Sum_{n>=0} q^(n*(n+1)/2) / (1-q)^n.

a(n) = Sum_{k=0..floor((sqrt(8*n+1)-3)/2)} C(n-1-C(k+1,2),k), for n >= 1.

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003
Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003
Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005
a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006
Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010
Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).
Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

			From _Gus Wiseman_, Sep 23 2019: (Start)
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (31)    (32)     (42)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (311)    (321)     (61)
                            (2111)   (411)     (421)
                            (11111)  (3111)    (511)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=1. - _N. J. A. Sloane_, Aug 31 2014
Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.

Cf. A067588, A090867, A116674, A173305.

Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.

f:=1/(1-x^2)/product(1-x^(2*j-1),j=1..32): fser:=series(f,x=0,62): seq(coeff(fser,x,n),n=0..58); # Emeric Deutsch, Feb 22 2006

mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)

# uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
print([a(n) for n in range(48)]) # Peter Luschny, Dec 04 2020

a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - Vladeta Jovovic, Sep 10 2003
a(n) = A000009(n) + a(n-2). - Vladeta Jovovic, Feb 10 2004
G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ (1/2) * A036469(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - Georg Fischer, Dec 04 2020

A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite multisets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  1   1   0   0   0   0   0   0   0   0   0   0
   n=3:  1   1   1   0   0   0   0   0   0   0   0   0
   n=4:  1   1   1   2   0   0   0   0   0   0   0   0
   n=5:  1   1   1   2   1   1   0   0   0   0   0   0
   n=6:  1   1   1   2   1   2   0   2   1   0   0   0
   n=7:  1   1   1   2   1   2   1   2   1   1   0   2
   n=8:  1   1   1   2   1   2   1   3   1   1   0   3
   n=9:  1   1   1   2   1   2   1   3   2   1   0   3
  n=10:  1   1   1   2   1   2   1   3   2   2   0   3
  n=11:  1   1   1   2   1   2   1   3   2   2   1   3
  n=12:  1   1   1   2   1   2   1   3   2   2   1   4
For example, the A(9,12) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 1
   n+k=4: 0 0 1 1
   n+k=5: 0 0 0 1 1
   n+k=6: 0 0 0 1 1 1
   n+k=7: 0 0 0 0 1 1 1
   n+k=8: 0 0 0 0 2 1 1 1
   n+k=9: 0 0 0 0 0 2 1 1 1
  n+k=10: 0 0 0 0 0 1 2 1 1 1
  n+k=11: 0 0 0 0 0 1 1 2 1 1 1
  n+k=12: 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=13: 0 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=14: 0 0 0 0 0 0 2 1 2 1 2 1 1 1
  n+k=15: 0 0 0 0 0 0 1 2 1 2 1 2 1 1 1
  n+k=16: 0 0 0 0 0 0 0 1 3 1 2 1 2 1 1 1
For example, antidiagonal n+k=10 counts the following partitions:
  n=5: (5)
  n=6: (411), (2211)
  n=7: (31111)
  n=8: (2111111)
  n=9: (111111111)
so the 10th antidiagonal is: (0,0,0,0,0,1,2,1,1,1).

Row sums are A000041 = partitions of n, strict A000009, no ones A002865.
Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
Antidiagonal sums are A379667.
The case without ones is A379668, antidiagonal sums A379669 (zeros A379670).
The strict case is A379671, antidiagonal sums A379672.
The strict case without ones is A379678, antidiagonal sums A379679 (zeros A379680).
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A096765, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

cp's OEIS Frontend

A002865 Number of partitions of n that do not contain 1 as a part.

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A087897 Number of partitions of n into odd parts greater than 1.

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A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

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A318029 Expansion of Sum_{k>=2} x^(k*(k+3)/2) / Product_{j=1..k} (1 - x^j).

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A379722 Numbers whose prime indices do not have the same sum as product.

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A307719 Number of partitions of n into 3 mutually coprime parts.

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A379736 Number of integer partitions of n whose product of parts is not n.

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