A000990 -id:A000990 - cp's OEIS frontend

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

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

A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 5, 3, 1, 1, 1, 5, 8, 10, 3, 1, 1, 1, 6, 14, 26, 16, 4, 1, 1, 1, 7, 20, 61, 60, 29, 4, 1, 1, 1, 8, 30, 128, 243, 184, 45, 5, 1, 1, 1, 9, 40, 254, 800, 1228, 488, 75, 5, 1, 1, 1, 10, 55, 467, 2518, 7252, 6684, 1509, 115, 6, 1

Offset: 0

Andrew Howroyd, Apr 08 2020

T(n,k) is the number of non-isomorphic n X n nonnegative integer symmetric matrices with all row and column sums equal to k and isomorphism being up to simultaneous permutation of rows and columns. The case that allows independent permutations of rows and columns is covered by A333737.

Terms may be computed without generating each graph by enumerating the graphs by degree sequence using dynamic programming. A PARI program showing this technique for the labeled case is given in A188403. Burnside's lemma as applied in A192517 can be used to extend this method to the unlabeled case.

			Array begins:
==============================================
n\k | 0 1  2   3    4     5      6       7
----+-----------------------------------------
  0 | 1 1  1   1    1     1      1       1 ...
  1 | 1 1  1   1    1     1      1       1 ...
  2 | 1 2  3   4    5     6      7       8 ...
  3 | 1 2  5   8   14    20     30      40 ...
  4 | 1 3 10  26   61   128    254     467 ...
  5 | 1 3 16  60  243   800   2518    6999 ...
  6 | 1 4 29 184 1228  7252  38194  175369 ...
  7 | 1 4 45 488 6684 78063 772243 6254652 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..377 (antidiagonals 0..26)

Rows n=0..4 are A000012, A000012, A000027(n+1), A006918(n+1), A333897.
Columns k=0..5 are A000012, A008619, A000990, A333894, A333895, A333896.
Cf. A188403, A192517, A263293, A333161, A333330, A333737, A334546.

A091356 Number of planar partitions of n with exactly 2 rows.

Original entry on oeis.org

1, 2, 5, 9, 18, 30, 53, 85, 139, 215, 336, 504, 760, 1115, 1635, 2351, 3375, 4770, 6725, 9368, 13006, 17885, 24510, 33319, 45139, 60743, 81457, 108610, 144334, 190844, 251542, 330082, 431825, 562710, 731154, 946644, 1222305, 1573155, 2019471

Offset: 2

Christian G. Bower, Jan 02 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Column 2 of A091355.

Cf. A000219, A091357, A091358, A091359, A091360.

b:= proc(n, k) option remember; `if`(n=0, 1, add(add(min(d, k)
     *d, d=numtheory[divisors](j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n, 2)-b(n, 1):
seq(a(n), n=2..50);  # Alois P. Heinz, Oct 02 2018

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[Sum[Min[d, k] d, {d, Divisors[j]}] b[n - j, k], {j, 1, n}]/n];
a[n_] :=  b[n, 2] - b[n, 1];
a /@ Range[2, 50] (* Jean-François Alcover, Oct 28 2020, after Alois P. Heinz *)

a(n) = A000990(n) - A000041(n).

A091357 Number of planar partitions of n with exactly 3 rows.

Original entry on oeis.org

1, 2, 5, 11, 22, 42, 78, 138, 239, 405, 669, 1088, 1741, 2744, 4267, 6564, 9975, 15019, 22394, 33111, 48549, 70678, 102127, 146636, 209186, 296697, 418401, 586985, 819218, 1137962, 1573336, 2165888, 2968914, 4053563, 5512820, 7469989

Offset: 3

Christian G. Bower, Jan 02 2004

nonn

Column 3 of A091355.

Cf. A000219, A089924, A091356-A091360.

a(n) = A000991(n)-A000990(n).

A093010 Triangle, read by rows, such that the convolution of the n-th row with the natural numbers forms the n-th diagonal, for n>=0, where each row begins with 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 7, 4, 1, 8, 14, 10, 5, 1, 10, 22, 22, 13, 6, 1, 12, 33, 40, 30, 16, 7, 1, 14, 45, 66, 58, 38, 19, 8, 1, 16, 60, 100, 104, 76, 46, 22, 9, 1, 18, 76, 146, 168, 142, 94, 54, 25, 10, 1, 20, 95, 202, 262, 242, 180, 112, 62, 28, 11, 1, 22, 115, 272, 386, 394, 316

Offset: 0

Paul D. Hanna, Mar 14 2004

Row sums form A000713, the number of partitions of n into parts of 3 kinds. Antidiagonal sums form A000990, the number of 2-line partitions of n.

			T(7,3) = 66 = 1*4+8*3+14*2+10*1 = T(4,0)*4+T(4,1)*3+T(4,2)*2+T(4,3)*1; this is also the third term of the 4th-diagonal.
The 6th antidiagonal is {1,10,14,4}, which has a sum of 29 = A000990(6) = number of 2-line partitions of 6.
Rows begin:
{1},
{1,2},
{1,4,3},
{1,6,7,4},
{1,8,14,10,5},
{1,10,22,22,13,6},
{1,12,33,40,30,16,7},
{1,14,45,66,58,38,19,8},
{1,16,60,100,104,76,46,22,9},
{1,18,76,146,168,142,94,54,25,10},
{1,20,95,202,262,242,180,112,62,28,11},
{1,22,115,272,386,394,316,218,130,70,31,12},...

Cf. A000713, A000990, A092905.

PARI
```
T(n,k)=if(n
				
```

T(n, k) = sum_{j=0..k} (k-j+1)*T(n-k, j), with T(0, n) = 1 for all n>=0.
A000713(n) = sum_{k=0..n} T(n, k) (row sums).
A000990(n) = sum_{k=0..floor(n/2)} T(n-k, k) (antidiagonal sums).

A147767 Triangle read by rows, square of A116598.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 5, 2, 2, 0, 1, 6, 5, 2, 2, 0, 1, 6, 5, 2, 2, 0, 1, 13, 6, 5, 2, 2, 0, 1, 16, 13, 6, 5, 2, 2, 0, 1, 30, 16, 13, 6, 5, 2, 2, 0, 1, 40, 30, 16, 13, 6, 5, 2, 2, 0, 1, 40, 30, 16, 13, 6, 5, 2, 2, 0, 1

Offset: 0

Gary W. Adamson, Nov 11 2008

Row sums = A000990: (1, 1, 3, 5, 10, 16, 29, 45, ...).

			First few rows of the triangle:
   1;
   0,  1;
   2,  0, 1;
   2,  2, 0, 1;
   5,  2, 2, 0, 1;
   6,  5, 2, 2, 0, 1;
  13,  6, 5, 2, 2, 0, 1;
  16, 13, 6, 5, 2, 2, 0, 1;
  ...

Cf. A147766, A000990.

Triangle read by rows, A116598^2. A147766 in every column.

Typo in formula corrected by Olivier Gérard, Jul 25 2016

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 4, 14, 65, 323, 1890, 12002, 83901, 630818, 5081318, 43546333, 395422430, 3788368227, 38151667046, 402516707510, 4436230390977, 50948789415297, 608433141666219, 7540823673023319, 96826154085714992, 1285991546051286085, 17640769457638701839, 249602608552024560609

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000990, A023871, A253289, A279215, A293554, A305206, A305653, A305655.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

A381265 a(n) is the number of solid (3D) partitions of n with 2 layers and second layer a plane partition of 3.

Original entry on oeis.org

6, 21, 57, 138, 294, 606, 1170, 2208, 4008, 7176, 12492, 21510, 36348, 60801, 100281, 164019, 265263, 425853

Offset: 3

Wouter Meeussen, Feb 18 2025

Conjecture: equal to 3*(2*A000219 -A000990 -2*A000041 +1) tested up to n=20.

			a(3)=6 since the 6 solid partitions of {3,3} are:
  z[{{3}},{{3}}],
  z[{{2,1}},{{2,1}}],
  z[{{1,1,1}},{{1,1,1}}],z[{{2},{1}},{{2},{1}}],
  z[{{1,1},{1}},{{1,1},{1}}],
  z[{{1},{1},{1}},{{1},{1},{1}}].

Wouter Meeussen, Mma functions for plane and solid partitions

Cf. A000041, A000219, A000990, A378854.

Table[Length@solidformBTK[{n,3}],{n,3,20}] (* or *)
g=20;3 CoefficientList[Series[2/Product[(1-x^m)^m,{m,g}]+ 1/(1-x)-(1-x)/Product[(1-x^m)^2,{m,g}]-2/Product[(1-x^m),{m,g}],{x,0,g}],x]

G.f.: 3*(2*Product_{k>0} 1/(1-x^k)^k -(1-x)*Product_{k>0} 1/(1-x^k)^2 - 2*Product_{k>0} 1/(1-x^k) + 1/(1 - x)) (conjectured).

cp's OEIS Frontend

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

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

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A333893 Array read by antidiagonals: T(n,k) is the number of unlabeled loopless multigraphs with n nodes of degree k or less.

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A091356 Number of planar partitions of n with exactly 2 rows.

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A091357 Number of planar partitions of n with exactly 3 rows.

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A093010 Triangle, read by rows, such that the convolution of the n-th row with the natural numbers forms the n-th diagonal, for n>=0, where each row begins with 1.

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A147767 Triangle read by rows, square of A116598.

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A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^n)).

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A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).