A000990 -id:A000990 - cp's OEIS frontend

A147766 Successive differences of A000990.

1, 0, 2, 2, 5, 6, 13, 16, 30, 40, 66, 90, 142, 192, 290, 396, 575, 782, 1112, 1500, 2092, 2808, 3848, 5132, 6945, 9192, 12298, 16178, 21422, 28000, 36763, 47748, 62205, 80334, 103910, 133458, 171538, 219150, 280039, 356020, 452469, 572548, 724047

Offset: 0

Gary W. Adamson, Nov 11 2008

nonn

Original definition: A000012^(-1) * A000990, where A000012^(-1) = the pairwise difference operator and A000990 = (1, 1, 3, 5, 10, 16, 29, 45, ...).

Triangle A147767 is constructed from A147766 in every column.

			A000990 = (1, 1, 3, 5, 10, 16, ...). Pairwise differences = (1, 0, 2, 2, 5, ...).

Cf. A000990, A147767.

a(n) = A000990(n) - A000990(n-1) and a(0)=1.
G.f.: exp(2*Sum_{k>=1} (sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(n) ~ exp(2*Pi*sqrt(n/3)) * Pi^2 / (4 * 3^(7/4) * n^(9/4)). - Vaclav Kotesovec, Aug 21 2018

Definition changed and more terms added by Olivier Gérard, Jul 25 2016

A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 195, 272, 373, 508, 684, 915, 1212, 1597, 2087, 2714, 3506, 4508, 5763, 7338, 9296, 11732, 14742, 18460, 23025, 28629, 35471, 43820, 53963, 66273, 81156, 99133, 120770, 146785, 177970, 215308, 259891, 313065, 376326, 451501

Offset: 0

N. J. A. Sloane

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004
With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).
Also, number of partitions of n into parts when there are two kinds of parts of size one.
Also number of graphical forest partitions of 2n+2.
a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004
Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005
Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005
Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008
Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008
Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008
Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009
a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010
a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011
Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012
With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012
a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013
a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015
a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016
a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018
a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example,  D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018
With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018
Also a(n) is the number of partitions of 2n+1 with exactly one odd part.
Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019
In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021
a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024
a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 45*x^7 + 67*x^8 + ...
From _Omar E. Pol_, Oct 25 2012: (Start)
For n = 5 consider the partitions of n+1:
--------------------------------------
.                         Number
Partitions of 6           of 1's
--------------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 0
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
35-16 =                     19
.
The difference between the sum of the first column and the sum of the second column of the set of partitions of 6 is 35 - 16 = 19 and equals the number of 1's in all partitions of 6, so the 6th term of this sequence is a(5) = 19.
(End)
From _Gus Wiseman_, Oct 26 2018: (Start)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose greatest part is > n:
  (2)  (4)   (6)    (8)     (A)      (C)
       (31)  (42)   (53)    (64)     (75)
             (51)   (62)    (73)     (84)
             (411)  (71)    (82)     (93)
                    (521)   (91)     (A2)
                    (611)   (622)    (B1)
                    (5111)  (631)    (732)
                            (721)    (741)
                            (811)    (822)
                            (6211)   (831)
                            (7111)   (921)
                            (61111)  (A11)
                                     (7221)
                                     (7311)
                                     (8211)
                                     (9111)
                                     (72111)
                                     (81111)
                                     (711111)
With offset 1, the a(1) = 1 through a(6) = 19 partitions of 2*n whose number of parts is > n:
  (11)  (211)   (2211)    (22211)     (222211)      (2222211)
        (1111)  (3111)    (32111)     (322111)      (3222111)
                (21111)   (41111)     (331111)      (3321111)
                (111111)  (221111)    (421111)      (4221111)
                          (311111)    (511111)      (4311111)
                          (2111111)   (2221111)     (5211111)
                          (11111111)  (3211111)     (6111111)
                                      (4111111)     (22221111)
                                      (22111111)    (32211111)
                                      (31111111)    (33111111)
                                      (211111111)   (42111111)
                                      (1111111111)  (51111111)
                                                    (222111111)
                                                    (321111111)
                                                    (411111111)
                                                    (2211111111)
                                                    (3111111111)
                                                    (21111111111)
                                                    (111111111111)
(End)
From _Joerg Arndt_, Jan 01 2024: (Start)
The a(5) = 19 multiset partitions of the multiset {1^5, 2^1} are:
   1:  {{1, 1, 1, 1, 1, 2}}
   2:  {{1, 1, 1, 1, 1}, {2}}
   3:  {{1, 1, 1, 1, 2}, {1}}
   4:  {{1, 1, 1, 1}, {1, 2}}
   5:  {{1, 1, 1, 1}, {1}, {2}}
   6:  {{1, 1, 1, 2}, {1, 1}}
   7:  {{1, 1, 1, 2}, {1}, {1}}
   8:  {{1, 1, 1}, {1, 1, 2}}
   9:  {{1, 1, 1}, {1, 1}, {2}}
  10:  {{1, 1, 1}, {1, 2}, {1}}
  11:  {{1, 1, 1}, {1}, {1}, {2}}
  12:  {{1, 1, 2}, {1, 1}, {1}}
  13:  {{1, 1, 2}, {1}, {1}, {1}}
  14:  {{1, 1}, {1, 1}, {1, 2}}
  15:  {{1, 1}, {1, 1}, {1}, {2}}
  16:  {{1, 1}, {1, 2}, {1}, {1}}
  17:  {{1, 1}, {1}, {1}, {1}, {2}}
  18:  {{1, 2}, {1}, {1}, {1}, {1}}
  19:  {{1}, {1}, {1}, {1}, {1}, {2}}
(End)

H. Gupta, An asymptotic formula in partitions. J. Indian Math. Soc., (N. S.) 10 (1946), 73-76.
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R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 6.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - N. J. A. Sloane, Dec 30 2018
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N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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T. D. Noe, Table of n, a(n) for n = 0..1000
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David Benson, Radha Kessar, and Markus Linckelmann, Hochschild cohomology of symmetric groups in low degrees, arXiv:2204.09970 [math.GR], 2022.
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L. Bracci and L. E. Picasso, A simple iterative method to write the terms of any order of perturbation theory in quantum mechanics, The European Physical Journal Plus, 127 (2012), Article 119. - From _N. J. A. Sloane_, Dec 31 2012
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D. Frank, C. D. Savage and J. A. Sellers, On the Number of Graphical Forest Partitions, preprint.
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M. D. Hirschhorn, The number of different parts in the partitions of n, Fib. Quart., 52 (2014), 10-15. See p. 11. - _N. J. A. Sloane_, Mar 25 2014
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N. J. A. Sloane, Transforms
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A diagonal of A066633.
Also second column of A126442. - George Beck, May 07 2011
Row sums of triangle A092905.
Also row sums of triangle A261555. - Omar E. Pol, Sep 14 2016
Also row sums of triangle A278427. - John P. McSorley, Nov 25 2016
Column k=2 of A292508.
Cf. A014153, A024786, A026794, A026905, A058884, A093694, A133735, A137633, A010815, A027293, A035363, A028310, A000712, A000990.
Cf. A000569, A025065, A036469, A096373, A147878, A209816, A320891, A320924.

List([0..45],n->Sum([0..n],k->NrPartitions(k))); # Muniru A Asiru, Jul 25 2018

a000070 = p a028310_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

with(combinat): a:=n->add(numbpart(j),j=0..n): seq(a(n), n=0..44); # Zerinvary Lajos, Aug 26 2008

CoefficientList[ Series[1/(1 - x)*Product[1/(1 - x^k), {k, 75}], {x, 0, 45}], x] (* Robert G. Wilson v, Jul 13 2004 *)
Table[ Count[ Flatten@ IntegerPartitions@ n, 1], {n, 45}] (* Robert G. Wilson v, Aug 06 2008 *)
Join[{1}, Accumulate[PartitionsP[Range[50]]]+1] (* _Harvey P. Dale, Mar 12 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (1 - x) / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 09 2013 *)
Accumulate[PartitionsP[Range[0,49]]] (* George Beck, Oct 23 2014; typo fixed by Virgile Andreani, Jul 10 2016 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(m=1, n, 1 - x^m, 1 + x * O(x^n)) / (1 - x), n))}; /* Michael Somos, Nov 08 2002 */

x='x+O('x^66); Vec(1/((1-x)*eta(x))) /* Joerg Arndt, May 15 2011 */

a(n) = sum(k=0, n, numbpart(k)); \\ Michel Marcus, Sep 16 2016

from itertools import accumulate
def A000070iter(n):
    L = [0]*n; L[0] = 1
    def numpart(n):
        S = 0; J = n-1; k = 2
        while 0 <= J:
            T = L[J]
            S = S+T if (k//2)%2 else S-T
            J -= k  if (k)%2 else k//2
            k += 1
        return S
    for j in range(1, n): L[j] = numpart(j)
    return accumulate(L)
print(list(A000070iter(100))) # Peter Luschny, Aug 30 2019

# Using function A365676Row. Compare also A365675.
from itertools import accumulate
def A000070List(size: int) -> list[int]:
    return [sum(accumulate(reversed(A365676Row(n)))) for n in range(size)]
print(A000070List(45))  # Peter Luschny, Sep 16 2023

def A000070_list(leng):
    p = [number_of_partitions(n) for n in range(leng)]
    return [add(p[:k+1]) for k in range(leng)]
A000070_list(45) # Peter Luschny, Sep 15 2014

Euler transform of [ 2, 1, 1, 1, 1, 1, 1, ...].
log(a(n)) ~ -3.3959 + 2.44613*sqrt(n). - Robert G. Wilson v, Jan 11 2002
a(n) = (1/n)*Sum_{k=1..n} (sigma(k)+1)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002
G.f.: (1/(1 - x))*Product_{m >= 1} 1/(1 - x^m).
a(n) seems to have the same parity as A027349(n+1). Comment from James Sellers, Mar 08 2006: that is true.
a(n) = A000041(2n+1) - A110618(2n+1) = A000041(2n+2) - A110618(2n+2). - Henry Bottomley, Aug 01 2005
Row sums of triangle A133735. - Gary W. Adamson, Sep 22 2007
a(n) = A092269(n+1) - A195820(n+1). - Omar E. Pol, Oct 20 2011
a(n) = A181187(n+1,1) - A181187(n+1,2). - Omar E. Pol, Oct 25 2012
From Peter Bala, Dec 23 2013: (Start)
Gupta gives the asymptotic result a(n-1) ~ sqrt(6/Pi^2)* sqrt(n)*p(n), where p(n) is the partition function A000041(n).
Let P(2,n) denote the set of partitions of n into parts k >= 2.
a(n-2) = Sum_{parts k in all partitions in P(2,n)} phi(k), where phi(k) is the Euler totient function (see A000010). Using this result and Mertens's theorem on the average order of the phi function, leads to the asymptotic result
a(n-2) ~ (6/Pi^2)*n*(p(n) - p(n-1)) = (6/Pi^2)*A138880(n) as n -> infinity. (End)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)) * (1 + 11*Pi/(24*sqrt(6*n)) + (73*Pi^2 - 1584)/(6912*n)). - Vaclav Kotesovec, Oct 26 2016
a(n) = A024786(n+2) + A024786(n+1). - Vaclav Kotesovec, Nov 05 2016
G.f.: exp(Sum_{k>=1} (sigma_1(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018
a(n) = A025065(2n). - Gus Wiseman, Oct 26 2018
a(n - 1) = A000041(2n) - A209816(n). - Gus Wiseman, Oct 26 2018

A000219 Number of plane partitions (or planar partitions) of n.

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658, 2483234, 3759612, 5668963, 8512309, 12733429, 18974973, 28175955, 41691046, 61484961, 90379784, 132441995, 193487501, 281846923

Offset: 0

N. J. A. Sloane

Two-dimensional partitions of n in which no row or column is longer than the one before it (compare A001970). E.g., a(4) = 13:
4.31.3.22.2.211.21..2.1111.111.11.11.1 but not 2
.....1....2.....1...1......1...11.1..1........ 11
....................1.............1..1
.....................................1
In the above, one also must require that rows & columns are nondecreasing, e.g., [1,1; 2] is also forbidden (which implies that row and column lengths are nondecreasing, if empty cells are identified with cells filled with 0's). - M. F. Hasler, Sep 22 2018
Can also be regarded as number of "safe pilings" of cubes in the corner of a room: the height should not increase away from the corner. - Wouter Meeussen
Also number of partitions of n objects of 2 colors, each part containing at least one black object; see example. - Christian G. Bower, Jan 08 2004
Number of partitions of n into 1 type of part 1, 2 types of part 2, ..., k types of part k. E.g., n=3 gives 111, 12, 12', 3, 3', 3''. - Jon Perry, May 27 2004
The bijection between the partitions in the two preceding comments goes by identifying a part with k black objects with a part of type k. - David Scambler and Joerg Arndt, May 01 2013
Can also be regarded as the number of Jordan canonical forms for an n X n matrix. (I.e., a 5 X 5 matrix has 24 distinct Jordan canonical forms, dependent on the algebraic and geometric multiplicity of each eigenvalue.) - Aaron Gable (agable(AT)hmc.edu), May 26 2009
(1/n) * convolution product of n terms * A001157 (sum of squares of divisors of n): (1, 5, 10, 21, 26, 50, 50, 85, ...) = a(n). As shown by [Bressoud, p. 12]: 1/6 * [1*24 + 5*13 + 10*6 + 21*3 + 26*1 + 50*1] = 288/6 = 48. - Gary W. Adamson, Jun 13 2009
Convolved with the aerated version (1, 0, 1, 0, 3, 0, 6, 0, 13, ...) = A026007: (1, 1, 2, 5, 8, 16, 28, 49, 83, ...). - Gary W. Adamson, Jun 13 2009
Starting with offset 1 = row sums of triangle A162453. - Gary W. Adamson, Jul 03 2009
Unfortunately, Wright's formula is also incomplete in the paper by G. Almkvist: "Asymptotic formulas and generalized Dedekind sums", p. 344, (the denominator should have sqrt(3*Pi) not sqrt(Pi)). This error was already corrected in the paper by Steven Finch: "Integer Partitions". - Vaclav Kotesovec, Aug 17 2015
Also the number of non-isomorphic weight-n chains of multisets whose dual is also a chain of multisets. The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. The weight of a multiset partition is the sum of sizes of its parts. - Gus Wiseman, Sep 25 2018

			A planar partition of 13:
  4 3 1 1
  2 1
  1
a(5) = (1/5!)*(sigma_2(1)^5+10*sigma_2(2)*sigma_2(1)^3+20*sigma_2(3)*sigma_2(1)^2+ 15*sigma_2(1)*sigma_2(2)^2+30*sigma_2(4)*sigma_2(1)+20*sigma_2(2)*sigma_2(3)+24*sigma_2(5)) = 24. - _Vladeta Jovovic_, Jan 10 2003
From _David Scambler_ and _Joerg Arndt_, May 01 2013: (Start)
There are a(4) = 13 partitions of 4 objects of 2 colors ('b' and 'w'), each part containing at least one black object:
1 black part:
  [ bwww ]
2 black parts:
  [ bbww ]
  [ bww, b ]
  [ bw, bw ]
3 black parts:
  [ bbbw ]
  [ bbw, b ]
  [ bb, bw ]
(but not: [bw, bb ] )
  [ bw, b, b ]
4 black parts:
  [ bbbb ]
  [ bbb, b ]
  [ bb, bb ]
  [ bb, b, b ]
  [ b, b, b, b ]
(End)
From _Geoffrey Critzer_, Nov 29 2014: (Start)
The corresponding partitions of the integer 4 are:
  4'''
  4''
  3'' + 1
  2' + 2'
  4'
  3' + 1
  2 + 2'
  2' + 1 + 1
  4
  3 + 1
  2 + 2
  2 + 1 + 1
  1 + 1 + 1 + 1.
(End)
From _Gus Wiseman_, Sep 25 2018: (Start)
Non-isomorphic representatives of the a(4) = 13 chains of multisets whose dual is also a chain of multisets:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{2},{1,2,2}}
  {{3},{1,2,3}}
  {{1,1},{1,1}}
  {{1,2},{1,2}}
  {{1},{1},{1,1}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}
(End)
G.f. = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 24*x^5 + 48*x^6 + 86*x^7 + 160*x^8 + ...

G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 241.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; pp(n) on p. 10.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.6).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.4.5).
P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Royal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - N. J. A. Sloane, May 21 2014
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).

Suresh Govindarajan, Table of n, a(n) for n = 0..6500 (first 401 terms from T. D. Noe)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
G. E. Andrews and P. Paule, MacMahon's partition analysis XII: Plane Partitions, J. Lond. Math. Soc., 76 (2007), 647-666.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
Michael Beeler, R. William Gosper and Richard C. Schroeppel, HAKMEM, ITEM 18, Memo AIM-239, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, Mass., 1972.
Edward A. Bender, Asymptotic methods in enumeration, SIAM Review 16 (1974), no. 4, p. 509.
E. A. Bender and D. E. Knuth, Enumeration of Plane Partitions, J. Combin. Theory A. 13, 40-54, 1972.
S. Benvenuti, B. Feng, A. Hanany and Y. H. He, Counting BPS operators in gauge theories: Quivers, syzygies and plethystics, arXiv:hep-th/0608050, p. 41-42.
Henry Bottomley, Illustration of initial terms
D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
Shouvik Datta, M. R. Gaberdiel, W. Li, and C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016.
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020.
Steven Finch, Integer Partitions, September 22, 2004. [Cached copy, with permission of the author]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 580.
Bernhard Heim, Markus Neuhauser and Robert Tröger, Inequalities for Plane Partitions, arXiv:2109.15145 [math.CO], 2021.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 141
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 18.
Vaclav Kotesovec, Graphs - The asymptotic ratio (250000 terms)
D. E. Knuth, A Note on Solid Partitions, Math. Comp. 24, 955-961, 1970.
Oleg Lazarev, Matt Mizuhara and Ben Reid, Some Results in Partitions, Plane Partitions, and Multipartitions, 13 August 2010.
P. A. MacMahon, Combinatory analysis.
J. Mangual, McMahon's Formula via Free Fermions, arXiv preprint arXiv:1210.7109 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 01 2013
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions ..., arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
L. Mutafchiev and E. Kamenov, On The Asymptotic Formula for the Number of Plane Partitions..., arXiv:math/0601253 [math.CO], 2006; C. R. Acad. Bulgare Sci. 59(2006), No. 4, 361-366.
Ken Ono, Sudhir Pujahari and Larry Rolen, Turán inequalities for the plane partition function, arXiv:2201.01352 [math.NT], 2022.
I. Pak, Partition bijections, a survey, Ramanujan J. 12 (2006) 5-75.
A. Rovenchak, Enumeration of plane partitions with a restricted number of parts, arXiv preprint arXiv:1401.4367 [math-ph], 2014.
Raphael Schumacher, The self-counting identity, Fib. Quart., 55 (No. 2 2017), 157-167.
N. J. A. Sloane, Transforms
J. Stienstra, Mahler measure, Eisenstein series and dimers, arXiv:math/0502197 [math.NT], 2005.
Balázs Szendrői, Non-commutative Donaldson-Thomas invariants and the conifold, Geometry & Topology 12.2 (2008): 1171-1202.
Eric Weisstein's World of Mathematics, Plane Partition
E. M. Wright, Rotatable partitions, J. London Math. Soc., 43 (1968), 501-505.
Index entries for "core" sequences

Cf. A000784, A000785, A000786, A005380, A005987, A048141, A048142, A089300.
Cf. A023871-A023878, A026007, A001157, A162453, A285216.
Differences: A191659, A191660, A191661.
Row sums of A089353 and A091438 and A091298.
Column k=1 of A144048. - Alois P. Heinz, Nov 02 2012
Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).
Cf. A249386, A249387.
Cf. A161870, A255610, A255611, A255612, A255613, A255614, A193427.

using Nemo, Memoize
@memoize function a(n)
    if n == 0 return 1 end
    s = sum(a(n - j) * divisor_sigma(j, 2) for j in 1:n)
    return div(s, n)
end
[a(n) for n in 0:20] # Peter Luschny, May 03 2020

series(mul((1-x^k)^(-k),k=1..64),x,63);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 17 2015

CoefficientList[Series[Product[(1 - x^k)^-k, {k, 64}], {x, 0, 64}], x]
Zeta[3]^(7/36)/2^(11/36)/Sqrt[3 Pi]/Glaisher E^(3 Zeta[3]^(1/3) (n/2)^(2/3) + 1/12)/n^(25/36) (* asymptotic formula after Wright; Vaclav Kotesovec, Jun 23 2014 *)
a[0] = 1; a[n_] := a[n] = Sum[a[n - j] DivisorSigma[2, j], {j, n}]/n; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
CoefficientList[Series[Exp[Sum[DivisorSigma[2, n] x^n/n, {n, 50}]], {x, 0, 50}], x] (* Eric W. Weisstein, Feb 01 2018 *)

{a(n) = if( n<0, 0, polcoeff( exp( sum( k=1, n, x^k / (1 - x^k)^2 / k, x * O(x^n))), n))}; /* Michael Somos, Jan 29 2005 */

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n))^-k), n))}; /* Michael Somos, Jan 29 2005 */

my(N=66, x='x+O('x^N)); Vec( prod(n=1,N, (1-x^n)^-n) ) \\ Joerg Arndt, Mar 25 2014

A000219(n)=#PlanePartitions(n) \\ See A091298 for PlanePartitions(). For illustrative use: much slower than the above. - M. F. Hasler, Sep 24 2018

from sympy import cacheit
from sympy.ntheory import divisor_sigma
@cacheit
def A000219(n):
    if n <= 1:
        return 1
    return sum(A000219(n - k) * divisor_sigma(k, 2) for k in range(1, n + 1)) // n
print([A000219(n) for n in range(20)])
# R. J. Mathar, Oct 18 2009

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n)
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{k >= 1} 1/(1 - x^k)^k. - MacMahon, 1912.
Euler transform of sequence [1, 2, 3, ...].
a(n) ~ (c_2 / n^(25/36)) * exp( c_1 * n^(2/3) ), where c_1 = A249387 = 2.00945... and c_2 = A249386 = 0.23151... - Wright, 1931. Corrected Jun 01 2010 by Rod Canfield - see Mutafchiev and Kamenov. The exact value of c_2 is e^(2c)*2^(-11/36)*zeta(3)^(7/36)*(3*Pi)^(-1/2), where c = Integral_{y=0..inf} (y*log(y)/(e^(2*Pi*y)-1))dy = (1/2)*zeta'(-1).
The exact value of c_1 is 3*2^(-2/3)*Zeta(3)^(1/3) = 2.0094456608770137530649... - Vaclav Kotesovec, Sep 14 2014
a(n) = (1/n) * Sum_{k=1..n} a(n-k)*sigma_2(k), n > 0, a(0)=1, where sigma_2(n) = A001157(n) = sum of squares of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: exp(Sum_{n>0} sigma_2(n)*x^n/n). a(n) = Sum_{pi} Product_{i=1..n} binomial(k(i)+i-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Jan 10 2003
From Vaclav Kotesovec, Nov 07 2016: (Start)
More precise asymptotics: a(n) ~ Zeta(3)^(7/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(11/36) * n^(25/36))
* (1 + c1/n^(2/3) + c2/n^(4/3) + c3/n^2), where
c1 = -0.23994424421250649114273759... = -277/(864*(2*Zeta(3))^(1/3)) - Zeta(3)^(2/3)/(1440*2^(1/3))
c2 = -0.02576771365117401620018082... = 353*Zeta(3)^(1/3)/(248832*2^(2/3)) - 17*Zeta(3)^(4/3)/(3225600*2^(2/3)) - 71575/(1492992*(2*Zeta(3))^(2/3))
c3 = -0.00533195302658826100834286... = -629557/859963392 - 42944125/(7739670528*Zeta(3)) + 14977*Zeta(3)/1114767360 - 22567*Zeta(3)^2/250822656000
and A = A074962 is the Glaisher-Kinkelin constant.
(End)

Corrected by N. J. A. Sloane, Jul 29 2006

Minor edits by Vaclav Kotesovec, Oct 27 2014

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

Original entry on oeis.org

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

A000991 Number of 3-line partitions of n.

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, 1274, 1983, 3032, 4610, 6915, 10324, 15235, 22371, 32554, 47119, 67689, 96763, 137404, 194211, 272939, 381872, 531576, 736923, 1016904, 1397853, 1913561, 2610023, 3546507, 4802694, 6481101, 8718309, 11689929, 15627591, 20828892

Offset: 0

N. J. A. Sloane

nonn

Planar partitions into at most three rows. - Joerg Arndt, May 01 2013

Number of partitions of n where there is one sort of part 1, two sorts of part 2, and three sorts of every other part. - Joerg Arndt, Mar 15 2014

L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.8).
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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000 (first 1000 terms from Alois P. Heinz)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.

A row of the array in A242641.

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^2*(1-x^2)/(&*[1-x^j: j in [1..2*m]])^3 )); // G. C. Greubel, Dec 06 2018

b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(min(i, 3)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[Min[i, 3]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[(1-x)^2 * (1-x^2) * Product[1/(1-x^k)^3, {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *)

x='x+O('x^66); Vec((1-x)^2*(1-x^2)/eta(x)^3) \\ Joerg Arndt, May 01 2013

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^2 * (1-x^2) / prod(1-x^j for j in (1..60))^3
s.coefficients()
# G. C. Greubel, Dec 06 2018

G.f.: (1-x)^2 * (1-x^2) / Product_(k>=1, 1-x^k )^3.
For n>=4, a(n) = A000716(n) - 2*A000716(n-1) + 2*A000716(n-3) - A000716(n-4). - Vaclav Kotesovec, Oct 28 2015
a(n) ~ Pi^3 * exp(Pi*sqrt(2*n)) / (16*n^3). - Vaclav Kotesovec, Oct 28 2015

G.f. corrected by Sean A. Irvine, Oct 19 2011
G.f. corrected by Joerg Arndt, May 01 2013
Prepended a(0)=1, added more terms, Joerg Arndt, May 01 2013

A002799 Number of 4-line partitions of n (i.e., planar partitions of n with at most 4 lines).

Original entry on oeis.org

1, 1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, 1794, 2882, 4544, 7131, 11031, 16983, 25844, 39124, 58680, 87538, 129578, 190830, 279140, 406334, 588026, 847034, 1213764, 1731780, 2459244, 3478185, 4898285, 6872041, 9603356, 13372607, 18553871, 25656865

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n where there is one sort of part 1, two sorts of part 2, three sorts of part 3, and four sorts of every other part. - Joerg Arndt, Mar 15 2014

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).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Alois P. Heinz)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
Vaclav Kotesovec, Graph - The asymptotic ratio (35000 terms)
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.
N. J. A. Sloane, Transforms

A row of the array in A242641.
Cf. A000219, A001452.
Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^3*(1-x^2)^2*(1-x^3)/(&*[1-x^j: j in [1..2*m]] )^4 )); // G. C. Greubel, Dec 06 2018

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> `if`(n<5,n,4)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[Min[#, 4]&]; Join[{1}, Table[a[n], {n, 1, 38}]] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[(1-x)^3 * (1-x^2)^2 * (1-x^3) * Product[1/(1-x^k)^4, {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *)

x='x+O('x^66); r=4; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r ) \\ Joerg Arndt, May 01 2013

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^3*(1-x^2)^2*(1-x^3)/prod(1-x^j for j in (1..60))^4
s.coefficients() # G. C. Greubel, Dec 06 2018

Euler transform of 1, 2, 3, 4, 4, 4, ...
G.f.: (1-x)^3 * (1-x^2)^2 * (1-x^3) / Product_{k>=1} (1-x^k)^4. - Joerg Arndt, May 01 2013
a(n) ~ 2^(13/4) * Pi^6 * exp(2*Pi*sqrt(2*n/3)) / (3^(13/4) * n^(19/4)). - Vaclav Kotesovec, Oct 28 2015

Edited and extended with formula by Christian G. Bower, Jan 01 2004
a(0)=1 prepended by Joerg Arndt, May 01 2013
Offset corrected by Vaclav Kotesovec, Oct 28 2015

A001452 Number of 5-line partitions of n.

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, 2118, 3462, 5564, 8888, 14016, 21973, 34081, 52552, 80331, 122078, 184161, 276303, 411870, 610818, 900721, 1321848, 1929981, 2805338, 4058812, 5847966, 8390097, 11990531, 17069145, 24210571, 34215537, 48190451, 67644522

Offset: 0

N. J. A. Sloane

nonn

Planar partitions into at most five rows. - Joerg Arndt, May 01 2013

Number of partitions of n where there are k sorts of parts k for k<=4 and 5 sorts all other parts. - Joerg Arndt, Mar 15 2014

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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (first 1000 terms from Alois P. Heinz)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms)
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.

A row of the array in A242641.

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/(&*[1-x^j: j in [1..2*m]])^5 )); // G. C. Greubel, Dec 06 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      min(d, 5)*d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 5]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[(1-x)^4 * (1-x^2)^3 * (1-x^3)^2 * (1-x^4) * Product[1/(1-x^k)^5, {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *)

x='x+O('x^66); r=5; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r ) \\ Joerg Arndt, May 01 2013

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)/prod(1-x^j for j in (1..60))^5
list(s) # G. C. Greubel, Dec 06 2018

G.f.: 1 / Product_{k>=1} (1-x^k)^min(k,5). - Sean A. Irvine, Jul 24 2012

a(n) ~ 15625 * Pi^10 * sqrt(5) * exp(Pi*sqrt(10*n/3)) / (2592 * sqrt(3) * n^7). - Vaclav Kotesovec, Oct 28 2015

More terms from Sean A. Irvine, Jul 24 2012

a(0)=1 prepended by Joerg Arndt, May 01 2013

A242641 Array read by antidiagonals upwards: B(s,n) ( s>=1, n >= 0) = number of s-line partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 3, 5, 5, 1, 1, 3, 6, 10, 7, 1, 1, 3, 6, 12, 16, 11, 1, 1, 3, 6, 13, 21, 29, 15, 1, 1, 3, 6, 13, 23, 40, 45, 22, 1, 1, 3, 6, 13, 24, 45, 67, 75, 30, 1, 1, 3, 6, 13, 24, 47, 78, 117, 115, 42, 1, 1, 3, 6, 13, 24, 48, 83, 141, 193, 181, 56, 1, 1, 3, 6, 13, 24, 48, 85, 152, 239, 319, 271, 77

Offset: 1

N. J. A. Sloane, May 21 2014

An s-line partition is a planar partition into at most s rows. s-line partitions of n are equinumerous with partitions of n with min(k,s) sorts of part k (cf. the g.f.). - Joerg Arndt, Feb 18 2015

Row s is asymptotic to (Product_{j=1..s-1} j!) * Pi^(s*(s-1)/2) * s^((s^2 + 1)/4) * exp(Pi*sqrt(2*n*s/3)) / (2^((s*(s+2)+5)/4) * 3^((s^2 + 1)/4) * n^((s^2 + 3)/4)). - Vaclav Kotesovec, Oct 28 2015

			Array begins:
1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
1, 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...
1, 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...
1, 1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...
1, 1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...
1, 1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...
1, 1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 281, 497, 851, 1460, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 499, 856, 1471, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 858, 1476, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1478, ...
1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, ...
...

Alois P. Heinz, Antidiagonals n = 1..200, flattened
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II.

Rows give A000041, A000990, A000991, A002799, A001452, A225196, A225197, A225198, A225199.
Main diagonal = A000219.
See A242642 for the upper triangle of the array.

# Maple code for the square array:
M:=100:
F:=s->mul((1-q^i)^(-i),i=1..s)*mul((1-q^j)^(-s),j=s+1..M);
A:=(s,n)->coeff(series(F(s),q,M),q,n);
for s from 1 to 12 do lprint( [seq(A(s,j),j=0..12)]); od:
# second Maple program:
B:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)
      *d, d=numtheory[divisors](j))*B(s, n-j), j=1..n)/n)
    end:
seq(seq(B(d-n, n), n=0..d-1), d=1..14);  # Alois P. Heinz, Oct 02 2018

M=100; F[s_] := Product[(1-q^i)^-i, {i, 1, s}]*Product[(1-q^j)^-s, {j, s+1, M}]; A[s_, n_] := Coefficient[Series[F[s], {q, 0, M}], q, n]; Table[A[s-j, j], {s, 1, 12}, {j, 0, s-1}] // Flatten (* Jean-François Alcover, Feb 18 2015, after Maple code *)

G.f. for row s: Product_{i=1..s} (1-q^i)^(-i) * Product_{j >= s+1} (1-q^j)^(-s). [MacMahon]

A225196 Number of 6-line partitions of n (i.e., planar partitions of n with at most 6 lines).

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, 2298, 3798, 6170, 9968, 15895, 25209, 39550, 61703, 95431, 146757, 224036, 340189, 513233, 770415, 1149933, 1708277, 2524846, 3715285, 5441762, 7937671, 11529512, 16681995, 24043245, 34527521, 49404590, 70452001, 100128249

Offset: 0

Joerg Arndt, May 01 2013

nonn

Number of partitions of n where there are k sorts of parts k for k<=5 and six sorts of all other parts. - Joerg Arndt, Mar 15 2014

Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms, convergence is slow)
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - _N. J. A. Sloane_, May 21 2014

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

A row of the array in A242641.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/(&*[1-x^j: j in [1..2*m]] )^6 )); // G. C. Greubel, Dec 06 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      min(d, 6)*d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 6]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Feb 18 2015, Alois P. Heinz *)
m:=50; CoefficientList[Series[(1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Product[(1-x^j), {j,1,m}])^6, {x,0,m}],x] (* G. C. Greubel, Dec 06 2018 *)

x='x+O('x^66); r=6; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )

R = PowerSeriesRing(ZZ,'x')
x = R.gen().O(50)
s = (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/prod(1-x^j for j in (1..60))^6
s.coefficients() # G. C. Greubel, Dec 06 2018

G.f.: 1/Product_{n>=1} (1-x^n)^min(n,6). - Joerg Arndt, Mar 15 2014
a(n) ~ 2160 * Pi^15 * exp(2*Pi*sqrt(n)) / n^(39/4). - Vaclav Kotesovec, Oct 28 2015
G.f.: (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Prod_{j>=1} (1-x^j ) )^6. - G. C. Greubel, Dec 06 2018

A225197 Number of 7-line partitions of n (i.e., planar partitions of n with at most 7 lines).

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, 2394, 3980, 6510, 10586, 17001, 27148, 42908, 67424, 105067, 162786, 250427, 383186, 582663, 881521, 1326319, 1986118, 2959376, 4390175, 6483255, 9534945, 13964910, 20374513, 29612085, 42883238, 61880879, 88993610, 127560266

Offset: 0

Joerg Arndt, May 01 2013

nonn

Number of partitions of n where there are k sorts of parts k for k<=6 and seven sorts of all other parts. - Joerg Arndt, Mar 15 2014

Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio (50000 terms, convergence is slow)
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table II. - _N. J. A. Sloane_, May 21 2014

A row of the array in A242641.

Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).

m:=50; r:=7; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^k)^(r-k): k in [1..r-1]])/(&*[1-x^j: j in [1..2*m]] )^r )); // G. C. Greubel, Dec 06 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      min(d, 7)*d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 7]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
m:=50; r:=7; CoefficientList[Series[Product[(1-x^k)^(r-k),{k,1,r-1}]/( Product[(1-x^j), {j,1,m}])^r, {x,0,m}],x] (* G. C. Greubel, Dec 06 2018 *)

x='x+O('x^66); r=7; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )

m=50; r=7
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(m)
s = prod((1-x^k)^(r-k) for k in (1..r-1))/prod(1-x^j for j in (1..m+2))^7
s.coefficients() # G. C. Greubel, Dec 06 2018

G.f.: 1/Product_{n>=1}(1-x^n)^min(n,7). - Joerg Arndt, Mar 15 2014

a(n) ~ 346032180025 * Pi^21 * sqrt(7) * exp(Pi*sqrt(14*n/3)) / (69984 * sqrt(3) * n^13). - Vaclav Kotesovec, Oct 28 2015

cp's OEIS Frontend

A147766 Successive differences of A000990.

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A000070 a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

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A000219 Number of plane partitions (or planar partitions) of n.

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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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A000991 Number of 3-line partitions of n.

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