A206240 -id:A206240 - cp's OEIS frontend

A206226 Number of partitions of n^2 into parts not greater than n.

1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - J. M. Bergot, Mar 26 2014 [computations done by Charles R Greathouse IV]
The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - Joerg Arndt, Mar 31 2014
From Vaclav Kotesovec, May 25 2015: (Start)
In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
-------
j  c(j)
1  0.1582087202672504149766310999238...
2  0.0794245035465730707705885572860...
3  0.0530017980244665552354063060738...
4  0.0397666338404544208556554596295...
5  0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
   c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j    d(j)
1    9.15337019245412246194853029240... = A258268
2   16.57962120993269533568313969522...
3   23.98280768122086592445663786762...
4   31.37931997386325137074644287711...
5   38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
   d(j) for big j asymptotically approaches j * exp(2).
(End)
d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - Vaclav Kotesovec, Jun 11 2015

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206227, A206240, A107379, A258268.
Column k=2 of A238016.
Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n^2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)
(* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* Vaclav Kotesovec, Jun 11 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
for(n=0,25,print1(a(n),", "))

a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014

A258268 Decimal expansion of a constant related to A206226.

Original entry on oeis.org

9, 1, 5, 3, 3, 7, 0, 1, 9, 2, 4, 5, 4, 1, 2, 2, 4, 6, 1, 9, 4, 8, 5, 3, 0, 2, 9, 2, 4, 0, 1, 3, 5, 4, 5, 4, 0, 0, 7, 3, 3, 2, 7, 2, 0, 4, 1, 2, 1, 8, 4, 8, 8, 4, 9, 6, 8, 9, 2, 6, 3, 2, 0, 1, 4, 7, 6, 1, 3, 8, 3, 7, 6, 6, 8, 9, 5, 7, 3, 1, 6, 2, 3, 9, 1, 5, 1, 9, 0, 2, 5, 5, 8, 7, 9, 5, 1, 9, 2, 8, 4, 5, 3, 8, 9

Offset: 1

Vaclav Kotesovec, May 25 2015

			9.153370192454122461948530292401354540073...

Cf. A206226, A206227, A206240, A258234, A107379, A173519, A238016, A258789.

r^3/(r-1) /.FindRoot[-PolyLog[2, 1-r] == 3*Log[r]^2/2, {r, E}, WorkingPrecision->120] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A206226(n)^(1/n).
Equals limit n->infinity A206227(n)^(1/n).
Equals limit n->infinity A206240(n)^(1/n).

More digits from Vaclav Kotesovec, Jun 10 2015

A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

Original entry on oeis.org

2, 9, 10, 68, 78, 98, 99, 105, 110, 125, 328, 444, 558, 620, 783, 812, 870, 966, 988, 1012, 1035, 1150, 1156, 1168, 1197, 1254, 1326, 1330, 1425, 1521, 1666, 1683, 1690, 1704, 1785, 1870, 1911, 2002, 2125, 2145, 2275, 2401, 2412, 2541, 2662, 2680, 2695, 3025

Offset: 1

Gus Wiseman, Jul 05 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
    2: {1}
    9: {2,2}
   10: {1,3}
   68: {1,1,7}
   78: {1,2,6}
   98: {1,4,4}
   99: {2,2,5}
  105: {2,3,4}
  110: {1,3,5}
  125: {3,3,3}
  328: {1,1,1,13}
  444: {1,1,2,12}
  558: {1,2,2,11}
  620: {1,1,3,11}
  783: {2,2,2,10}
  812: {1,1,4,10}
  870: {1,2,3,10}
  966: {1,2,4,9}
  988: {1,1,6,8}

Partitions of this type are counted by A364055, without  zeros A206240.
The RHS is A001222.
The LHS is A326567/A326568.
A008284 counts partitions by length, A058398 by mean.
A088529/A088530 gives mean of prime signature A124010.
A112798 lists prime indices, sum A056239.
A124943 counts partitions by low median, high A124944.
A316413 ranks partitions with integer mean, counted by A067538.
A326622 counts factorizations with integer mean, strict A328966.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A025065, A327473, A327476, A327482, A359889, A363723, A363727, A363729, A363944, A363949.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Mean[prix[#]]==PrimeOmega[#]&]

A304176 Number of partitions of n^3 into exactly n parts.

Original entry on oeis.org

1, 1, 4, 61, 1906, 91606, 6023602, 505853354, 51900711796, 6306147384659, 886745696653253, 141778041323736643, 25417656781153090889, 5052180112449982704619, 1103058286595668300801794, 262487324530101028337614478, 67628783852463631751658038290

Offset: 0

Seiichi Manyama, May 07 2018

nonn

			n | Partitions of n^3 into exactly n parts
--+------------------------------------------------------------
1 |   1.
2 |   7+1 = 6+2 = 5+3 = 4+4.
3 |   25+ 1+1 = 24+ 2+1 = 23+ 3+1 = 23+ 2+2 = 22+ 4+1 = 22+ 3+2
  | = 21+ 5+1 = 21+ 4+2 = 21+ 3+3 = 20+ 6+1 = 20+ 5+2 = 20+ 4+3
  | = 19+ 7+1 = 19+ 6+2 = 19+ 5+3 = 19+ 4+4 = 18+ 8+1 = 18+ 7+2
  | = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2 = 17+ 7+3 = 17+ 6+4
  | = 17+ 5+5 = 16+10+1 = 16+ 9+2 = 16+ 8+3 = 16+ 7+4 = 16+ 6+5
  | = 15+11+1 = 15+10+2 = 15+ 9+3 = 15+ 8+4 = 15+ 7+5 = 15+ 6+6
  | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6
  | = 13+13+1 = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6
  | = 13+ 7+7 = 12+12+3 = 12+11+4 = 12+10+5 = 12+ 9+6 = 12+ 8+7
  | = 11+11+5 = 11+10+6 = 11+ 9+7 = 11+ 8+8 = 10+10+7 = 10+ 9+8
  | =  9+ 9+9.

Chai Wah Wu, Table of n, a(n) for n = 0..136 (terms 0..96 from Alois P. Heinz)

Cf. A206240, A238608, A304208, A304212.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+b(n-i, min(i, n-i)))
    end:
a:= n-> b(n^3-n, n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 07 2018

$RecursionLimit = 2000;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + b[n - i, Min[i, n - i]]];
a[n_] := b[n^3 - n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

{a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^3-n)))), n^3-n)}

import sys
from functools import lru_cache
sys.setrecursionlimit(10**6)
@lru_cache(maxsize=None)
def b(n,i): return 1 if n == 0 or i == 1 else b(n,i-1)+b(n-i,min(i,n-i))
def A304176(n): return b(n**3-n,n) # Chai Wah Wu, Sep 09 2021, after Alois P. Heinz

a(n) = [x^(n^3-n)] Product_{k=1..n} 1/(1-x^k).

A206227 Number of partitions of n^2+n into parts not greater than n.

Original entry on oeis.org

1, 1, 4, 19, 108, 674, 4494, 31275, 225132, 1662894, 12541802, 96225037, 748935563, 5900502806, 46976736513, 377425326138, 3056671009814, 24930725879856, 204623068332997, 1688980598900228, 14012122025369431, 116784468316023069, 977437078888272796, 8212186058546599006

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206226, A206240, A107379, A258268.

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2+n, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2+n)))),n^2+n)}
for(n=0,30,print1(a(n),", "))

a(n) = [x^(n^2+n)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.3572966225745094270279188015952797... . - Vaclav Kotesovec, Sep 07 2014

A347585 Number of partitions of n^2 into n or more parts.

Original entry on oeis.org

1, 1, 4, 25, 201, 1773, 16751, 165083, 1681341, 17562238, 187255089, 2030853040, 22344663465, 248900855994, 2802367768848, 31848644363490, 364960085991118, 4212964989100093, 48953036382441044, 572178690287957687, 6723501191850208483, 79388206896842420091

Offset: 0

Seiichi Manyama, Sep 08 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A058399, A072213, A161408, A206226, A206240.

a(n) = [x^(n^2)] Sum_{k>=n} x^k / Product_{j=1..k} (1 - x^j).
a(n) = A072213(n) + A206240(n) - A206226(n).
a(n) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2). - Vaclav Kotesovec, Sep 14 2021

A361800 Number of integer partitions of n with the same length as median.

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 1, 2, 3, 3, 3, 3, 4, 6, 9, 13, 14, 15, 18, 21, 27, 32, 40, 46, 55, 62, 72, 82, 95, 111, 131, 157, 186, 225, 264, 316, 366, 430, 495, 578, 663, 768, 880, 1011, 1151, 1316, 1489, 1690, 1910, 2158, 2432, 2751, 3100, 3505, 3964, 4486, 5079, 5764

Offset: 1

Gus Wiseman, Apr 07 2023

nonn

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

			The a(1) = 1 through a(15) = 9 partitions (A=10, B=11):
  1  .  .  22  .  .  331  332  333  433  533  633  733   833   933
           31             431  432  532  632  732  832   932   A32
                               531  631  731  831  931   A31   B31
                                                   4441  4442  4443
                                                         5441  5442
                                                         5531  5532
                                                               6441
                                                               6531
                                                               6621

For minimum instead of median we have A006141, for twice minimum A237757.
For maximum instead of median we have A047993, for twice length A237753.
For maximum instead of length we have A053263, for twice median A361849.
For mean instead of median we have A206240 (zeros removed).
For minimum instead of length we have A361860.
For twice median we have A362049, ranks A362050.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices.
Cf. A008284, A013580, A027193, A079309, A240219, A362048.

Table[Length[Select[IntegerPartitions[n],Length[#]==Median[#]&]],{n,30}]

A347606 Number of partitions of n^n into exactly n parts.

Original entry on oeis.org

0, 1, 2, 61, 117874, 33219689231, 2559960025059106420, 85975912953927216830024650654, 1841153609473379088124269084031755459049386

Offset: 0

Seiichi Manyama, Sep 08 2021

Cf. A206240, A304176, A347605, A347607.

a(n) = polcoef(prod(k=1, n, 1/(1-x^k+x*O(x^(n^n-n)))), n^n-n);

a(n) = [x^(n^n-n)] Product_{k=1..n} 1/(1-x^k).

A347617 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k into exactly n parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 4, 7, 1, 0, 1, 1, 8, 61, 34, 1, 0, 1, 1, 16, 547, 1906, 192, 1, 0, 1, 1, 32, 4921, 117874, 91606, 1206, 1, 0, 1, 1, 64, 44287, 7478386, 53830967, 6023602, 8033, 1, 0, 1, 1, 128, 398581, 477568114, 33219689231, 43054503928, 505853354, 55974, 1, 0

Offset: 0

Seiichi Manyama, Sep 08 2021

			Square array begins:
  0, 1,   1,     1,        1,           1, ...
  1, 1,   1,     1,        1,           1, ...
  0, 1,   2,     4,        8,          16, ...
  0, 1,   7,    61,      547,        4921, ...
  0, 1,  34,  1906,   117874,     7478386, ...
  0, 1, 192, 91606, 53830967, 33219689231, ...

Columns k=0..3 give A063524, A000012, A206240, A304176.
Main diagonal gives A347606.
Cf. A238016, A347615, A347618.

T(n, k) = if(k==0, n==1, polcoef(prod(j=1, n, 1/(1-x^j+x*O(x^(n^k-n)))), n^k-n));

T(n,k) = [x^(n^k-n)] Product_{j=1..n} 1/(1-x^j).

A364055 Number of integer partitions of n satisfying (length) = (mean). Partitions of n into sqrt(n) parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0, 192, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1206, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8033, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55974, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 07 2023

nonn

			The a(0) = 1 through a(9) = 7 partitions:
  ()  (1)  .  .  (22)  .  .  .  .  (333)
                 (31)              (432)
                                   (441)
                                   (522)
                                   (531)
                                   (621)
                                   (711)

The strict case is A107379(sqrt(n)).
Without  zeros we have A206240.
These partitions have ranks A363951.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean, ranks A316413.
Cf. A025065, A026905, A237984, A327472, A327482, A363944, A363949.

Table[Length[If[n==0,{{}},Select[IntegerPartitions[n],Mean[#]==Length[#]&]]],{n,0,30}]

a(n^2) = A206240(n).

cp's OEIS Frontend

A206226 Number of partitions of n^2 into parts not greater than n.

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A258268 Decimal expansion of a constant related to A206226.

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A363951 Numbers whose prime indices satisfy (length) = (mean), or (sum) = (length)^2.

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A304176 Number of partitions of n^3 into exactly n parts.

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A206227 Number of partitions of n^2+n into parts not greater than n.

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A347585 Number of partitions of n^2 into n or more parts.

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A361800 Number of integer partitions of n with the same length as median.

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A347606 Number of partitions of n^n into exactly n parts.

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