A144064 -id:A144064 - cp's OEIS frontend

A000712 Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds.

1, 2, 5, 10, 20, 36, 65, 110, 185, 300, 481, 752, 1165, 1770, 2665, 3956, 5822, 8470, 12230, 17490, 24842, 35002, 49010, 68150, 94235, 129512, 177087, 240840, 326015, 439190, 589128, 786814, 1046705, 1386930, 1831065, 2408658, 3157789, 4126070, 5374390

Offset: 0

N. J. A. Sloane

For n >= 1, a(n) is also the number of conjugacy classes in the automorphism group of the n-dimensional hypercube. This automorphism group is the wreath product of the cyclic group C_2 and the symmetric group S_n, its order is in sequence A000165. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 04 2001
Also, number of noncongruent matrices in GL_n(Z): each Jordan block can only have +1 or -1 on the diagonal. - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
a(n) = Sum (k(1)+1)*(k(2)+1)*...*(k(n)+1), where the sum is taken over all (k(1),k(2),...,k(n)) such that k(1)+2*k(2)+...+n*k(n) = n, k(i)>=0, i=1..n, cf. A104510, A077285. - Vladeta Jovovic, Apr 21 2005
Convolution of partition numbers (A000041) with itself. - Graeme McRae, Jun 07 2006
Number of one-to-one partial endofunctions on n unlabeled points. Connected components are either cycles or "lines", hence two for each size. - Franklin T. Adams-Watters, Dec 28 2006
Equals A000716: (1, 3, 9, 22, 561, 108, ...) convolved with A010815. A000716 = the number of partitions of n into parts of 3 kinds = the Euler transform of [3,3,3,...]. - Gary W. Adamson, Oct 26 2008
Paraphrasing the g.f.: 1 + 2x + 5x^2 + ... = s(x) * s(x^2) * s(x^3) * s(x^4) * ...; where s(x) = 1 + 2x + 3x^2 + 4x^3 + ... is (up to a factor x) the g.f. of A000027. - Gary W. Adamson, Apr 01 2010
Also equals number of partitions of 2n in which the odd parts appear as many times in even as in odd positions. - Wouter Meeussen, Apr 17 2013
Also number of ordered pairs (R,S) with R a partition of r, S a partition of s, and r+s=n; see example.  This corresponds to the formula a(n) = sum(r+s==n, p(r)*p(s) ) = Sum_{k=0..n} p(k)*p(n-k). - Joerg Arndt, Apr 29 2013
Also the number of all multi-graphs with exactly n-edges and with vertex degrees 1 or 2. - Ebrahim Ghorbani, Dec 02 2013
If one decomposes k-permutations into cycles and so-called paths, the number of different type of decompositions equals to a(k); see the paper by Chen, Ghorbani, and Wong. - Ebrahim Ghorbani, Dec 02 2013
Let T(n,k) be the number of partitions of n having parts 1 through k of two kinds, with T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-2,2) + T(n-3,3) + ... - Gregory L. Simay, May 18 2019
Also the number of orbits of projections in the partition monoid P_n under conjugation by permutations. - James East, Jul 21 2020

			Assume there are integers of two kinds: k and k'; then a(3) = 10 since 3 has the following partitions into parts of two kinds: 111, 111', 11'1', 1'1'1', 12, 1'2, 12', 1'2', 3, and 3'. - _W. Edwin Clark_, Jun 24 2011
There are a(4)=20 partitions of 4 into 2 sorts of parts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:0  1:1  ]
03:  [ 1:0  1:0  1:1  1:1  ]
04:  [ 1:0  1:1  1:1  1:1  ]
05:  [ 1:1  1:1  1:1  1:1  ]
06:  [ 2:0  1:0  1:0  ]
07:  [ 2:0  1:0  1:1  ]
08:  [ 2:0  1:1  1:1  ]
09:  [ 2:0  2:0  ]
10:  [ 2:0  2:1  ]
11:  [ 2:1  1:0  1:0  ]
12:  [ 2:1  1:0  1:1  ]
13:  [ 2:1  1:1  1:1  ]
14:  [ 2:1  2:1  ]
15:  [ 3:0  1:0  ]
16:  [ 3:0  1:1  ]
17:  [ 3:1  1:0  ]
18:  [ 3:1  1:1  ]
19:  [ 4:0  ]
20:  [ 4:1  ]
- _Joerg Arndt_, Apr 28 2013
The a(4)=20 ordered pairs (R,S) of partitions for n=4 are
  ([4], [])
  ([3, 1], [])
  ([2, 2], [])
  ([2, 1, 1], [])
  ([1, 1, 1, 1], [])
  ([3], [1])
  ([2, 1], [1])
  ([1, 1, 1], [1])
  ([2], [2])
  ([2], [1, 1])
  ([1, 1], [2])
  ([1, 1], [1, 1])
  ([1], [3])
  ([1], [2, 1])
  ([1], [1, 1, 1])
  ([], [4])
  ([], [3, 1])
  ([], [2, 2])
  ([], [2, 1, 1])
  ([], [1, 1, 1, 1])
This list was created with the Sage command
   for P in PartitionTuples(2,4) : print P;
- _Joerg Arndt_, Apr 29 2013
G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 + 185*x^8 + ...

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Proposition 2.5.2 on page 78.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)
Arvind Ayyer, Amritanshu Prasad, and Steven Spallone, Macdonald trees and determinants of representations for finite Coxeter groups, arXiv:1812.00608 [math.RT], 2018.
M. Baake, Structure and representations of the hyperoctahedral group, J. Math. Phys. 25 (1984) 3171, table 1.
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
Jan Brandts and A Cihangir, Enumeration and investigation of acute 0/1-simplices modulo the action of the hyperoctahedral group, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
E. R. Canfield, C. D. Savage and H. S. Wilf, Regularly spaced subsums of integer partitions, arXiv:math/0308061 [math.CO], 2003.
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Hilbert Space Fragmentation in the Chiral Luttinger Liquid, arXiv:2409.10359 [cond-mat.str-el], 2024. See pp. 8, 11.
Alexandre Chaduteau, Nyan Raess, Henry Davenport, and Frank Schindler, Momentum-space modulated symmetries in the Luttinger liquid, Phys. Rev. B (2025) Vol. 111, Art. No. 165105. See p. 9.
B. F. Chen, E. Ghorbani, and K. B. Wong, Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs, Electronic J. Combin. 20(4) (2013), #P22.
W. Y. C. Chen, K. Q. Ji and H. S. Wilf, BG-ranks and 2-cores, arXiv:math/0605474 [math.CO], 2006.
W. Edwin Clark, Mohamed Elhamdadi, Xiang-dong Hou, Masahico Saito and Timothy Yeatman, Connected Quandles Associated with Pointed Abelian Groups, arXiv preprint arXiv:1107.5777 [math.RA], 2011.
W. Edwin Clark and Xiang-dong Hou, Galkin Quandles, Pointed Abelian Groups, and Sequence A000712 arXiv:1108.2215 [math.CO], Aug 10, 2011. [added by Jonathan Vos Post]
Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
M. De Salvo, D. Fasino, D. Freni, and G. Lo Faro, Fully simple semihypergroups, transitive digraphs, and sequence A000712, Journal of Algebra, Volume 415, 1 October 2014, pp. 65-87.
Mario De Salvo, Dario Fasino, Domenico Freni, and Giovanni Lo Faro, Semihypergroups Obtained by Merging of 0-semigroups with Groups, Filomat (2018) Vol. 32, No. 12, 4177-4194.
Paul Hammond and Richard Lewis, Congruences in ordered pairs of partitions, Int. J. Math. Math. Sci. (2004), no.45--48, 2509--2512.
Ruth Hoffmann, Özgür Akgün, and Christopher Jefferson, Composable Constraint Models for Permutation Enumeration, arXiv:2311.17581 [cs.DM], 2023.
Saud Hussein, An Identity for the Partition Function Involving Parts of k Different Magnitudes, arXiv:1806.05416 [math.NT], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 129.
Han Mao Kiah, Anshoo Tandon, and Mehul Motani, Generalized Sphere-Packing Bound for Subblock-Constrained Codes, arXiv:1901.00387 [cs.IT], 2019.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
Yen-chi R. Lin and Shu-Yen Pan, A recursive relation for bipartition numbers, arXiv:2406.14851 [math.CO], 2024.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and A. M. Läuchli, Chiral spin liquids in triangular lattice SU (N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
Sylvain Prolhac, Spectrum of the totally asymmetric simple exclusion process on a periodic lattice--first excited states, arXiv preprint arXiv:1404.1315 [cond-mat.stat-mech], 2014.
N. J. A. Sloane, Transforms.
Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000165, A000041, A002107 (reciprocal of g.f.).
Cf. A002720.
Cf. A000716, A010815. - Gary W. Adamson, Oct 26 2008
Row sums of A175012. - Gary W. Adamson, Apr 03 2010
Column k=2 of A144064.
Cf. A008619, A000070, A000990, A285217, A304045.

a000712 = p a008619_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012

# DedekindEta is defined in A000594.
A000712List(len) = DedekindEta(len, -2)
A000712List(39) |> println # Peter Luschny, Mar 09 2018

with(combinat): A000712:= n-> add(numbpart(k)*numbpart(n-k), k=0..n): seq(A000712(n), n=0..40); # Emeric Deutsch

CoefficientList[ Series[ Product[1/(1 - x^n)^2, {n, 40}], {x, 0, 37}], x]; (* Robert G. Wilson v, Feb 03 2005 *)
Table[Count[Partitions[2*n], q_ /; Tr[(-1)^Mod[Flatten[Position[q, ?OddQ]], 2]] === 0], {n, 12}] (* _Wouter Meeussen, Apr 17 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^-2, {x, 0, n}]; (* Michael Somos, Oct 12 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@n], {n, 0, 15}] (* Robert Price, Jun 15 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / eta(x + A)^2, n))}; /* Michael Somos, Nov 14 2002 */

Vec(1/eta('x+O('x^66))^2) /* Joerg Arndt, Jun 25 2011 */

from sympy import npartitions
def A000712(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) # Chai Wah Wu, Sep 25 2023

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1, 2, 2)
b = EulerTransform(a)
print([b(n) for n in range(40)]) # Peter Luschny, Nov 11 2020

a(n) = Sum_{k=0..n} p(k)*p(n-k), where p(n) = A000041(n).
Euler transform of period 1 sequence [ 2, 2, 2, ...]. - Michael Somos, Jul 22 2003
a(n) = A006330(n) + A001523(n). - Michael Somos, Jul 22 2003
a(0) = 1, a(n) = (1/n)*Sum_{k=0..n-1} 2*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
a(n) ~ (1/12)*3^(1/4)*n^(-5/4)*exp((2/3)*sqrt(3)*Pi*sqrt(n)). - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
G.f.: Product_{i>=1} (1 + x^i)^(2*A001511(i)) (see A000041). - Jon Perry, Jun 06 2004
More precise asymptotics: a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/(12*sqrt(3)) + 15*sqrt(3)/(16*Pi)) / sqrt(n) + (Pi^2/864 + 315/(512*Pi^2) + 35/192)/n). - Vaclav Kotesovec, Jan 22 2017
From Peter Bala, Jan 26 2016: (Start)
a(n) is odd iff n = 2*m and p(m) is odd.
a(n) = (2/n)*Sum_{k = 0..n} k*p(k)*p(n-k) for n >= 1.
Conjecture: : a(n) is divisible by 5 when n is congruent to 2, 3 or 4 modulo 5. (End)
Conjecture is proved in Hammond and Lewis. - Yen-chi R. Lin, Jun 24 2024
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018
With the convention that a(n) = 0 for n < 0 we have the recurrence a(n) = g(n) + Sum_{k >= 1} (-1)^(k+1)*(2*k + 1)*a(n - k*(k + 1)/2), where g(n) = (-1)^m if n = m*(3*m - 1)/2 is a generalized pentagonal number (A001318) else g(n) = 0. For example, n = 7 = -2*(3*(-2) - 1)/2 is a pentagonal number, g(7) = 1, and so a(7) = 1 + 3*a(6) - 5*a(4) + 7*a(1) = 1 + 195 - 100 + 14 = 110. - Peter Bala, Apr 06 2022
a(n) = p(n/2) + Sum_{k \in Z, k != 0} (-1)^{k-1} a(n-k^2), here p(n) = A000041(n) and p(x) = 0 when x is not an integer. - Yen-chi R. Lin, Jun 24 2024
Conjecture: a(25*n + 23) is divisible by 25 (checked for n < 400). - Peter Bala, Jan 13 2025

More terms from Joe Keane (jgk(AT)jgk.org), Nov 17 2001
More terms from Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 15 2004
Definition rewritten by N. J. A. Sloane, Apr 02 2022

A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

Original entry on oeis.org

0, 1, 3, 7, 15, 29, 54, 95, 163, 270, 439, 696, 1088, 1669, 2530, 3780, 5591, 8173, 11845, 17000, 24215, 34210, 48008, 66895, 92660, 127554, 174651, 237830, 322297, 434625, 583524, 779972, 1038356, 1376787, 1818755, 2393775, 3139812, 4104433, 5348375, 6947545, 8998201, 11620313, 14965126, 19220569

Offset: 0

Thomas Wieder, Sep 11 2006

nonn

Consider a fragmentation process of an n-object which consists of n unlabeled elements (= 1-parts). By definition the n-object can scatter into up to n m-parts where an m-part consists of 1 up to n elements. A 4-object can split up for example into 4 1-parts which corresponds to the integer partition [1,1,1,1], or it can, for example, rest unfragmented which corresponds to [4]. Since the number of integer partitions of n=4 equals 5, there are 5 n=4-fragmentation processes.
Now we ask for the probability of getting an m-part after an n-fragmentation. Think of a Greek statue which had been broken into n parts and covered by earth. We could find several m-parts, in the most lucky case we would find all m-parts which add up to m_1+m_2+...+m_n=n. Then the statue could be restored.
For example for n=4 we could ask for the probability prob(n=4,m=2) of just a single 2-part. We have 2 cases for a 2-part and we have 15 cases in total, thus prob(n=4,m=2)=2/15 (the 2 cases come from [1,1,2] and [2,2]). The chances to find the two 2-parts from the [2,2]-fragmentation are 1/15 only. The chances to find the n=4-object unsplitted are also 1/15 only.
This sequence is generated over the unordered partitions; for example, when n = 4 there are 1+3+2+5+4 = 15 cases. If we allow a null case for each of the five partitions then we have 15+5 = 20 which is A000712(4). - Alford Arnold, Dec 12 2006
Number of partitions into two kinds of parts with the first kind of parts used in each partition. - Joerg Arndt, Jun 21 2011

			a(n=4) = 15 because the possible combinations of all five integer partitions of n=4 are: [1], [1, 1], [1, 1, 1], [1, 1, 1, 1], [1], [2], [1, 1], [1, 2], [1, 1, 2], [2], [2, 2], [1], [3], [1, 3], [4].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000041, A000079, A000262.
Cf. A000712, A074139, A074141.
Cf. A048574, A144064.

a122768 n = a122768_list !! n
a122768_list = 0 : f (tail a000041_list) [1] where
   f (p:ps) rs = (sum $ zipWith (*) rs $ tail a000041_list) : f ps (p : rs)
-- Reinhard Zumkeller, Nov 09 2015

A122768 := proc(n::integer) local i,j,prttnlst,prttn,ZahlTeile,H; prttnlst:=partition(n); H := NULL; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; ZahlTeile := nops(prttn); for j from 1 to ZahlTeile do H := H,op(choose(prttn,j)); od; od; print(n,H,nops([H])); end proc;
A000712 := proc(n) option remember ; add(combinat[numbpart](k)*combinat[numbpart](n-k),k=0..n) ; end: A000041 := proc(n) combinat[numbpart](n) ; end: A122768 := proc(n::integer) RETURN( A000712(n)-A000041(n)) ; end: for n from 0 to 80 do printf("%d,",A122768(n)) ; od: # R. J. Mathar, Aug 25 2008
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, add(
      k*numtheory[sigma](j)*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n,2)-b(n,1):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 31 2017

1/QPochhammer[x]^2 - 1/QPochhammer[x] + O[x]^50 // CoefficientList[#, x]& (* Jean-François Alcover, Feb 05 2017, after Joerg Arndt *)

x='x+O('x^66); /* that many terms */
Vec(1/eta(x)^2-1/eta(x)) /* show terms (omitting initial zero) */
/* Joerg Arndt, Jun 21 2011 */

from sympy import npartitions
def A122768(n): return (sum(npartitions(k)*npartitions(n-k) for k in range(1,n+1>>1))<<1) + (0 if n&1 else npartitions(n>>1)**2) + npartitions(n) if n else 0 # Chai Wah Wu, Sep 25 2023

G.f.: 1/P(x)^2 - 1/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011
With sum_i^P(n) = the sum over all P(n) integer partitions of n, sum_j^p(i) = the sum over all p(i) parts of the i-th integer partition, prttn(i) = the i-th partition whereat prttn(i) is a list, choose(L,k) = construct the list LC of combinations of a list L (see Maple), |LC| = number of elements of list LC (=Maple's nops command) we have a(n) = sum_i^P(n) sum_j^p(i) |choose(prttn,j)|
a(n) = A000712(n) - A000041(n). - Alford Arnold, Dec 12 2006
a(n) = A144064(n,2)-A144064(n,1). - Alois P. Heinz, Mar 31 2017
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(3/4)*n^(5/4)) * (1 - (Pi/12 + 45/(16*Pi))/sqrt(3*n)). - Vaclav Kotesovec, Mar 31 2017

Extended by R. J. Mathar, Aug 25 2008

A000716 Number of partitions of n into parts of 3 kinds.

Original entry on oeis.org

1, 3, 9, 22, 51, 108, 221, 429, 810, 1479, 2640, 4599, 7868, 13209, 21843, 35581, 57222, 90882, 142769, 221910, 341649, 521196, 788460, 1183221, 1762462, 2606604, 3829437, 5590110, 8111346, 11701998, 16790136, 23964594, 34034391, 48104069, 67679109, 94800537, 132230021, 183686994, 254170332

Offset: 0

N. J. A. Sloane

nonn

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 122.
Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table I.
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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from T. D. Noe)
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, 2016, hal-01285685v2.
Victor J. W. Guo and Jiang Zeng, Two truncated identities of Gauss, arXiv 1205.4340 [math.CO], 2012. - _N. J. A. Sloane_, Oct 10 2012
Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 391
Vladimir P. Kostov, Asymptotic expansions of zeros of a partial theta function, arXiv:1504.00883 [math.CA], 2015.
Vladimir P. Kostov, Stabilization of the asymptotic expansions of the zeros of a partial theta function, arXiv preprint arXiv:1510.02584 [math.CA], 2015.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
P. Nataf, M. Lajkó, A. Wietek, K. Penc, F. Mila, and A. M. Läuchli, Chiral spin liquids in triangular lattice SU(N) fermionic Mott insulators with artificial gauge fields, arXiv preprint arXiv:1601.00958 [cond-mat.quant-gas], 2016.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000712.

Column 3 of A144064.

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*3, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, May 20 2013

a[0] = 1; a[n_] := a[n] = 1/n*Sum[3*a[k]*DivisorSigma[1, n-k], {k, 0, n-1}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 03 2014, after Joerg Arndt *)
(1/QPochhammer[q]^3 + O[q]^40)[[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

Vec(1/eta('x+O('x^66))^3) \\ Joerg Arndt, Apr 28 2013

from functools import lru_cache
from sympy import divisor_sigma
@lru_cache(maxsize=None)
def A000716(n): return sum(A000716(k)*divisor_sigma(n-k) for k in range(n))*3//n if n else 1 # Chai Wah Wu, Sep 25 2023

G.f.: Product_{m>=1} 1/(1-x^m)^3.
EULER transform of 3, 3, 3, 3, 3, 3, 3, 3, ...
a(0)=1, a(n) = 1/n*Sum_{k=0..n-1} 3*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
a(n) ~ exp(Pi * sqrt(2*n)) / (8 * sqrt(2) * n^(3/2)) * (1 - (3/Pi + Pi/8) / sqrt(2*n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

Extended with formula from Christian G. Bower, Apr 15 1998

A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 2, 0, 1, 4, 6, 6, 2, 0, 1, 5, 10, 13, 9, 3, 0, 1, 6, 15, 24, 24, 14, 4, 0, 1, 7, 21, 40, 51, 42, 22, 5, 0, 1, 8, 28, 62, 95, 100, 73, 32, 6, 0, 1, 9, 36, 91, 162, 206, 190, 120, 46, 8, 0, 1, 10, 45, 128, 259, 384, 425, 344, 192, 66, 10, 0

Offset: 0

Ilya Gutkovskiy, May 07 2017

A(n,k) is the number of partitions of n into distinct parts (or odd parts) with k types of each part.

			A(3,2) = 6 because we have [3], [3'], [2, 1], [2', 1], [2, 1'] and [2', 1'] (partitions of 3 into distinct parts with 2 types of each part).
Also A(3,2) = 6 because we have [3], [3'], [1, 1, 1], [1, 1, 1'], [1, 1', 1'] and [1', 1', 1'] (partitions of 3 into odd parts with 2 types of each part).
Square array begins:
  1,  1,  1,   1,   1,   1,  ...
  0,  1,  2,   3,   4,   5,  ...
  0,  1,  3,   6,  10,  15,  ...
  0,  2,  6,  13,  24,  40,  ...
  0,  2,  9,  24,  51,  95,  ...
  0,  3, 14,  42, 100, 206,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
N. J. A. Sloane, Transforms

Columns k=0-32 give: A000007, A000009, A022567-A022596.
Rows n=0-2 give: A000012, A001477, A000217.
Main diagonal gives A270913.
Antidiagonal sums give A299106.
Cf. A144064, A286352, A308680.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
     (t-> b(t, min(t, i-1), k)*binomial(k, j))(n-i*j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 29 2019

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A308680(n,k-i). - Alois P. Heinz, Aug 29 2019

A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

Original entry on oeis.org

1, 1, 5, 22, 105, 506, 2492, 12405, 62337, 315445, 1605340, 8207563, 42124380, 216903064, 1119974875, 5796944357, 30068145905, 156250892610, 813310723925, 4239676354650, 22130265931900, 115654632452535, 605081974091875, 3168828466966388, 16610409114771900

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

nonn

Number of partitions of n into parts of n kinds. - Vladeta Jovovic, Sep 08 2002
Main diagonal of A144064. - Omar E. Pol, Jun 27 2012
From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p >= 3. Cf. A270913. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000712, A000716, A023003-A023021, A005758, A006922.

Cf. A109085, A192435, A252782, A255526, A255672, A270913, A270915, A270919.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= n-> etr(j->n)(n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008

a[n_] := SeriesCoefficient[ Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Feb 24 2015 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2016 *)
Table[SeriesCoefficient[Exp[n*Sum[x^j/(j*(1-x^j)), {j, 1, n}]], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}

{a(n)=n*polcoeff(log(1/x*serreverse(x*eta(x+x*O(x^n)))), n)} /* Paul D. Hanna, Apr 05 2012 */

a(n) = Sum_{pi} Product_{i=1..n} binomial(k_i+n-1, k_i) where pi runs through all nonnegative solutions of k_1+2*k_2+...+n*k_n=n. a(n) = b(n, n) where b(n, m)= m/n*Sum_{i=1..n} sigma(i)*b(n-i, m) is recurrence for number of partitions of n into parts of m kinds. - Vladeta Jovovic, Sep 08 2002
Equals the logarithmic derivative of A109085, the g.f. of which is (1/x)*Series_Reversion(x*eta(x)). - Paul D. Hanna, Apr 05 2012
Let G(x) = exp( Sum_{n>=1} a(n)*x^n/n ), then G(x) = 1/Product_{n>=1} (1-x^n*G(x)^n) is the g.f. of A109085. - Paul D. Hanna, Apr 05 2012
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165..., c = A327279 = 0.26801521271073331568695383828... . - Vaclav Kotesovec, Sep 10 2014

a(0)=1 prepended by Alois P. Heinz, Mar 30 2015

A246935 Number A(n,k) of partitions of n into k sorts of parts; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 14, 5, 0, 1, 5, 20, 39, 34, 7, 0, 1, 6, 30, 84, 129, 74, 11, 0, 1, 7, 42, 155, 356, 399, 166, 15, 0, 1, 8, 56, 258, 805, 1444, 1245, 350, 22, 0, 1, 9, 72, 399, 1590, 4055, 5876, 3783, 746, 30, 0

Offset: 0

Alois P. Heinz, Sep 08 2014

In general, column k > 1 is asymptotic to c * k^n, where c = Product_{j>=1} 1/(1-1/k^j) = 1/QPochhammer[1/k,1/k]. - Vaclav Kotesovec, Mar 19 2015

When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field of size k. - Geoffrey Critzer, Nov 11 2022

			A(2,2) = 6: [2a], [2b], [1a,1a], [1a,1b], [1b,1a], [1b,1b].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,      1,      1, ...
  0,  1,   2,    3,     4,      5,      6,      7, ...
  0,  2,   6,   12,    20,     30,     42,     56, ...
  0,  3,  14,   39,    84,    155,    258,    399, ...
  0,  5,  34,  129,   356,    805,   1590,   2849, ...
  0,  7,  74,  399,  1444,   4055,   9582,  19999, ...
  0, 11, 166, 1245,  5876,  20455,  57786, 140441, ...
  0, 15, 350, 3783, 23604, 102455, 347010, 983535, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Columns k=0-10 give: A000007, A000041, A070933, A242587, A246936, A246937, A246938, A246939, A246940, A246941, A246942.
Rows n=0-4 give: A000012, A001477, A002378, A027444, A186636.
Main diagonal gives A124577.
Cf. A144064, A255970, A256130.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]]; A[n_, k_] := b[n, n, k];  Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 03 2015, after Alois P. Heinz *)

G.f. of column k: Product_{i>=1} 1/(1-k*x^i).

T(n,k) = Sum_{i=0..k} C(k,i) * A255970(n,i).

A067687 Expansion of 1/( 1 - x / Product_{n>=1} (1-x^n) ).

Original entry on oeis.org

1, 1, 2, 5, 12, 29, 69, 165, 393, 937, 2233, 5322, 12683, 30227, 72037, 171680, 409151, 975097, 2323870, 5538294, 13198973, 31456058, 74966710, 178662171, 425791279, 1014754341, 2418382956, 5763538903, 13735781840, 32735391558, 78015643589

Offset: 0

Alford Arnold, Feb 05 2002

nonn

Previous name was: Invert transform of right-shifted partition function (A000041).
Sums of the antidiagonals of the array formed by sequences A000007, A000041, A000712, A000716, ... or its transpose A000012, A000027, A000096, A006503, A006504, ....
Row sums of triangle A143866 = (1, 2, 5, 12, 29, 69, 165, ...) and right border of A143866 = (1, 1, 2, 5, 12, ...). - Gary W. Adamson, Sep 04 2008
Starting with offset 1 = A137682 / A000041; i.e. (1, 3, 7, 17, 40, 96, ...) / (1, 2, 3, 5, 7, 11, ...). - Gary W. Adamson, May 01 2009
From L. Edson Jeffery, Mar 16 2011: (Start)
Another approach is the following. Let T be the infinite lower triangular matrix with columns C_k (k=0,1,2,...) such that C_0=A000041 and, for k > 0, such that C_k is the sequence giving the number of partitions of n into parts of k+1 kinds (successive self-convolutions of A000041 yielding A000712, A000716, ...) and shifted down by k rows. Then T begins (ignoring trailing zero entries in the rows)
  (1,  0, ...            )
  (1,  1, 0, ...         )
  (2,  2, 1, 0, ...      )
  (3,  5, 3, 1, 0, ...   )
  (5, 10, 9, 4, 1, 0, ...)
etc., and a(n) is the sum of entries in row n of T. (End)

			The array begins:
  1,  1,  1,   1,   1,  1,  1, 1, ...
  0,  1,  2,   3,   4,  5,  6, 7, ...
  0,  2,  5,   9,  14, 20, 27, ...
  0,  3, 10,  22,  40, 65, ...
  0,  5, 20,  51, 105, ...
  0,  7, 36, 108, ...
  0, 11, 65, ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.
N. J. A. Sloane, Transforms

Cf. A000007, A000041, A000712, A000716, A000012, A000027, A000096, A006503, A006504.
Cf. table A060850.
Cf. A137682, A143866.
Antidiagonal sums of A144064.

N=66; x='x+O('x^N); et=eta(x); Vec( sum(n=0,N, x^n/et^n ) ) \\ Joerg Arndt, May 08 2009

a(n) = Sum_{k=1..n} A000041(k-1)*a(n-k). - Vladeta Jovovic, Apr 07 2003
O.g.f.: 1/(1-x*P(x)), P(x) - o.g.f. for number of partitions (A000041). - Vladimir Kruchinin, Aug 10 2010
a(n) ~ c / r^n, where r = A347968 = 0.419600352598356478498775753566700025318... is the root of the equation QPochhammer(r) = r and c = 0.3777957165566422058901624844315414446044096308877617181754... = Log[r]/(Log[(1 - r)*r] + QPolyGamma[1, r] - Log[r]*Derivative[0, 1][QPochhammer][r, r]). - Vaclav Kotesovec, Feb 16 2017, updated Mar 31 2018

More terms from Vladeta Jovovic, Apr 07 2003
More terms and better definition from Franklin T. Adams-Watters, Mar 14 2006
New name (using g.f. by Vladimir Kruchinin), Joerg Arndt, Feb 19 2014

A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 22, 21, 8, 1, 11, 43, 59, 36, 10, 1, 15, 80, 144, 124, 55, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1

Offset: 1

Alford Arnold, Apr 16 2001

Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022

			Table begins:
   1;
   2,   1;
   3,   4,    1;
   5,  10,    6,    1;
   7,  22,   21,    8,    1;
  11,  43,   59,   36,   10,    1;
  15,  80,  144,  124,   55,   12,   1;
  22, 141,  321,  362,  225,   78,  14,   1;
  30, 240,  669,  944,  765,  370, 105,  16,  1;
  42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
  ...
For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=1-10 give: A000041, A048574, A341221, A341222, A341223, A341225, A341226, A341227, A341228, A341236.
Row sums give A055887.
Cf. A097805, A010815, A055888, A144064, A261719, A326346.
T(2n,n) gives A340987.

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 12 2015
# Uses function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, combinat:-numbpart); # Peter Luschny, Oct 07 2022

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - Alois P. Heinz, Mar 12 2015
Sum_{k=1..n} k * T(n,k) = A326346(n). - Alois P. Heinz, Sep 11 2019
Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - Alois P. Heinz, Feb 07 2021
G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021

More terms from Vladeta Jovovic, Jan 02 2004

cp's OEIS Frontend

A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

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A000712 Generating function = Product_{m>=1} 1/(1 - x^m)^2; a(n) = number of partitions of n into parts of 2 kinds.

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A122768 Number of combinations which can be taken from the integer partitions of n. Total number of cases in the (n,m)-fragmentation process.

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A000716 Number of partitions of n into parts of 3 kinds.

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A286335 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)^k.

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A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

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