A242641 -id:A242641 - cp's OEIS frontend

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

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

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

A225199 Number of 9-line partitions of n (i.e., planar partitions of n with at most 9 lines).

Original entry on oeis.org

1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 499, 856, 1471, 2466, 4124, 6788, 11110, 17965, 28890, 45995, 72819, 114354, 178577, 276952, 427279, 655199, 999773, 1517388, 2292377, 3446462, 5159352, 7689517, 11414606, 16875813, 24856366, 36474188, 53334376, 77717219, 112874158, 163403202

Offset: 0

Joerg Arndt, May 01 2013

nonn

Number of partitions of n where there are k sorts of parts k for k<=8 and nine sorts of all other parts. - Joerg Arndt, Mar 15 2014

In general, "number of r-line partitions" is asymptotic to (Product_{j=1..r-1} j!) * Pi^(r*(r-1)/2) * r^((r^2 + 1)/4) * exp(Pi*sqrt(2*n*r/3)) / (2^((r*(r+2)+5)/4) * 3^((r^2 + 1)/4) * n^((r^2 + 3)/4)). - Vaclav Kotesovec, Oct 28 2015

Vincenzo Librandi and Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000
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, p. 28.
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:=9; 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

b:= proc(n,i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(min(i, 9)+j-1, j)*
       b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 15 2014

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 9]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
m:=50; r:=9; 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=9; Vec( prod(k=1,r-1, (1-x^k)^(r-k)) / eta(x)^r )

m=50; r=9
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,9). - Joerg Arndt, Mar 15 2014

a(n) ~ 2101805306799541875 * sqrt(3) * Pi^36 * exp(Pi*sqrt(6*n)) / (8*n^21). [The convergence is very slow, numerical verification needs more than 1000000 terms.] - Vaclav Kotesovec, Oct 28 2015

A242643 a(n) = Sum_{d|n, d <= 6} d^2 + 6*Sum_{d|n, d>6} d.

Original entry on oeis.org

1, 5, 10, 21, 26, 50, 43, 69, 64, 90, 67, 138, 79, 131, 125, 165, 103, 212, 115, 226, 178, 203, 139, 330, 176, 239, 226, 315, 175, 405, 187, 357, 274, 311, 278, 516, 223, 347, 322, 514, 247, 554, 259, 483, 449, 419, 283, 714, 337, 540, 418, 567, 319, 698, 422, 699, 466, 527, 355, 973, 367, 563, 610, 741, 494, 842

Offset: 1

N. J. A. Sloane, May 21 2014

nonn

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 I. Note that the entry 53 should be 50.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A row of the array in A242641.

sd6[n_]:=Module[{d=Divisors[n]},Total[Select[d,#<7&]^2]+6Total[Select[ d,#>6&]]];Array[sd6,70] (* Harvey P. Dale, Mar 18 2015 *)

Typo in definition corrected by N. J. A. Sloane, Mar 18 2015

A242642 Triangle read by rows: T(s,n) (n>=1, 1 <= s <= n) = number of s-line partitions of n.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 21 2014

			Triangle begins:
[1]
[2, 3]
[3, 5, 6]
[5, 10, 12, 13]
[7, 16, 21, 23, 24]
[11, 29, 40, 45, 47, 48]
[15, 45, 67, 78, 83, 85, 86]
[22, 75, 117, 141, 152, 157, 159, 160]
...
The square array (A242641 with n=0 column omitted) begins:
1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, ...
1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, ...
1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, ...
1, 3, 6, 13, 23, 45, 78, 141, 239, 409, 674, 1116, ...
1, 3, 6, 13, 24, 47, 83, 152, 263, 457, 768, 1292, ...
1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, ...
1, 3, 6, 13, 24, 48, 86, 159, 279, 492, 840, 1436, ...
...

Alois P. Heinz, Rows 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.

Upper triangle of array in A242641 (with the n=0 column omitted).

T:= proc(s, n) option remember; `if`(n=0, 1, add(add(min(d, s)
      *d, d=numtheory[divisors](j))*T(s, n-j), j=1..n)/n)
    end:
seq(seq(T(s, n), s=1..n), n=1..14);  # Alois P. Heinz, Oct 02 2018

T[s_, n_] := T[s, n] = If[n==0, 1, Sum[Sum[Min[d, s]*d, {d, Divisors[j]}]* T[s, n - j], {j, 1, n}]/n];
Table[Table[T[s, n], {s, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 10 2019, after Alois P. Heinz *)

cp's OEIS Frontend

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

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A000991 Number of 3-line partitions of n.

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

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A001452 Number of 5-line partitions of n.

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

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