A225199 -id:A225199 - cp's OEIS frontend

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

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

A000990 Number of plane partitions of n with at most two rows.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, 605, 895, 1291, 1866, 2648, 3760, 5260, 7352, 10160, 14008, 19140, 26085, 35277, 47575, 63753, 85175, 113175, 149938, 197686, 259891, 340225, 444135, 577593, 749131, 968281, 1248320, 1604340, 2056809, 2629357, 3353404

Offset: 0

N. J. A. Sloane

Equals row sums of triangle A147767. - Gary W. Adamson, Nov 11 2008
Also number of partitions of n into parts of 2 kinds except for 1. - Reinhard Zumkeller, Nov 06 2012
Antidiagonal sums of triangle A093010.

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 105.
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. (1.7).
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, Table of n, a(n) for n = 0..10000 (terms n = 1001..7000 from Vaclav Kotesovec)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
M. S. Cheema, Letter to N. J. A. Sloane, Jul 15 1970 [scanned copy]
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.
R. Newton and A. R. Camacho, Strangely dual orbifold equivalence I, arXiv preprint arXiv:1509.08069 [math.QA], 2015.
Steven Rayan, Aspects of the topology and combinatorics of Higgs bundle moduli spaces, arXiv:1809.05732 [math.AG], 2018.
A. G. Shannon, B. Kuloğlu, and E. Özkan, Rhaly terraced sequences their generalizations, properties and applications, Comp. Appl. Math. 44, 226 (2025). See p. 2.

A row of the array in A242641.
Cf. A147767. - Gary W. Adamson, Nov 11 2008
Cf. A008619, A000070, A000712.
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).

a000990 = p $ tail 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

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

b:= proc(n,i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(min(i, 2)+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, 2]+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 17 2014, after Alois P. Heinz *)
Flatten[{1, Differences[Table[Sum[PartitionsP[j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 50}]]}] (* Vaclav Kotesovec, Oct 28 2015 *)
CoefficientList[(1-q)/QPochhammer[q]^2+O[q]^50, q] (* Jean-François Alcover, Nov 27 2015 *)

a(n)=if(n<0,0,polcoeff((1-x)/prod(k=1,n,1-x^k,1+x*O(x^n))^2,n)) /* Michael Somos, Jan 29 2005 */

{a(n)=polcoeff(exp(sum(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))*x^m/m)),n)} \\ Paul D. Hanna, Apr 22 2010

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

s=((1-x)/prod(1-x^j for j in (1..60))^2).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018

G.f.: 1 / ( (1-x) * Product_{m>=2} (1-x^m)^2 ) = (1-x) / Product_{m>=1} (1-x^m)^2.
G.f.: exp( Sum_{n>=1} ((1+x^n)/(1-x^n))*x^n/n ). - Paul D. Hanna, Apr 22 2010
For n>=1, a(n) = A000712(n) - A000712(n-1). - Vaclav Kotesovec, Oct 28 2015
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 28 2015
G.f.: exp(Sum_{k>=1} (2*sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018

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

A225198 Number of 8-line partitions of n (i.e., planar partitions of n with at most 8 lines).

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 48, 86, 160, 281, 497, 851, 1460, 2442, 4076, 6692, 10928, 17623, 28266, 44873, 70842, 110910, 172674, 266942, 410512, 627387, 954113, 1443063, 2172456, 3254446, 4854236, 7208018, 10659872, 15700111, 23035956, 33671399, 49042600, 71179250, 102963936, 148452294

Offset: 0

Joerg Arndt, May 01 2013

nonn

Number of partitions of n where there are k sorts of parts k for k<=7 and eight 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
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
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms, convergence is very slow)

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:=8; 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 10 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      min(d, 8)*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, 8]*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:=8; 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 10 2018 *)

x='x+O('x^66); r=8; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )

m=50; r=8
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))^r
s.coefficients() # G. C. Greubel, Dec 10 2018

G.f.: 1/Product_{n>=1}(1-x^n)^min(n,8). - Joerg Arndt, Mar 15 2014

a(n) ~ 7696581394432000 * sqrt(2) * Pi^28 * exp(4*Pi*sqrt(n/3)) / (19683 * 3^(1/4) * n^(67/4)). - Vaclav Kotesovec, Oct 28 2015

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Julia

Maple

Mathematica

PARI

PARI

PARI

PARI

Python

SageMath

Formula

Extensions

A000990 Number of plane partitions of n with at most two rows.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Sage

Formula

A000991 Number of 3-line partitions of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A001452 Number of 5-line partitions of n.

Views

Author

Keywords

Comments

References

Links

Crossrefs