A072243 -id:A072243 - cp's OEIS frontend

A072213 Number of partitions of n^2.

1, 1, 5, 30, 231, 1958, 17977, 173525, 1741630, 18004327, 190569292, 2056148051, 22540654445, 250438925115, 2814570987591, 31946390696157, 365749566870782, 4219388528587095, 49005643635237875, 572612058898037559

Offset: 0

Jeff Burch, Jul 03 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..900 (terms 0..250 from Alois P. Heinz)

Cf. A000041, A000290, A072243, A219430.

Cf. A161407, A161408. - Reinhard Zumkeller, Jun 10 2009

A072213 := proc(n) combinat[numbpart](n^2) ; end proc:
seq(A072213(n),n=0..10) ; # R. J. Mathar, Jan 24 2011

Mathematica
```
Table[ PartitionsP[n^2], {n, 1, 20}]
```
PARI
```
a(n)=numbpart(n^2)
```
PARI
```
a(n)=polcoeff(1/eta(x),n^2,x)
```

[number_of_partitions(n^2)for n in range(0,26)] # Zerinvary Lajos, Nov 26 2009

a(n) = A000041(n^2).

a(n) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2). - Vaclav Kotesovec, Dec 01 2015

A107379 Number of ways to write n^2 as the sum of n odd numbers, disregarding order.

Original entry on oeis.org

1, 1, 1, 3, 9, 30, 110, 436, 1801, 7657, 33401, 148847, 674585, 3100410, 14422567, 67792847, 321546251, 1537241148, 7400926549, 35854579015, 174677578889, 855312650751, 4207291811538, 20782253017825, 103048079556241, 512753419159803, 2559639388956793

Offset: 0

David Radcliffe, Sep 25 2009

Motivated by the fact that the n-th square is equal to the sum of the first n odd numbers.
Also the number of partitions of n^2 into n distinct parts. a(3) = 3: [1,2,6], [1,3,5], [2,3,4]. - Alois P. Heinz, Jan 20 2011
Also the number of partitions of n*(n-1)/2 into parts not greater than n. - Paul D. Hanna, Feb 05 2012
Also the number of partitions of n*(n+1)/2 into n parts. - J. Stauduhar, Sep 05 2017
Also the number of fair dice with n sides and expected value (n+1)/2 with distinct composition of positive integers. - Felix Huber, Aug 11 2024

			For example, 9 can be written as a sum of three odd numbers in 3 ways: 1+1+7, 1+3+5 and 3+3+3.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (first 200 terms from Alois P. Heinz)

Cf. A072243, A152140, A258191, A258192, A258234, A281489.

f := proc (n, k) option remember;
if n = 0 and k = 0 then return 1 end if;
if n <= 0 or n < k then return 0 end if;
if `mod`(n+k, 2) = 1 then return 0 end if;
if k = 1 then return 1 end if;
return procname(n-1, k-1) + procname(n-2*k, k)
end proc;
seq(f(k^2,k), k=0..20);

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n-1)/2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n*(n-1)/2)))),n*(n-1)/2)} /* Paul D. Hanna, Feb 05 2012 */

a(n) = A008284((n^2+n)/2,n) = A008284(A000217(n),n). - Max Alekseyev, Sep 25 2009
a(n) = [x^(n*(n-1)/2)] Product_{k=1..n} 1/(1 - x^k). - Paul D. Hanna, Feb 05 2012
a(n) ~ c * d^n / n^2, where d = 5.400871904118154152466091119104270052029... = A258234, c = 0.155212227152682180502977404265024265... . - Vaclav Kotesovec, Sep 07 2014

Arguments in the Maple program swapped and 4 terms added by R. J. Mathar, Oct 02 2009

A281489 Number of partitions of n^2 into distinct odd parts.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 33, 93, 276, 833, 2574, 8057, 25565, 81889, 264703, 861889, 2824974, 9311875, 30851395, 102676439, 343112116, 1150785092, 3872588051, 13071583810, 44245023261, 150145281903, 510721124972, 1741020966255, 5947081503460, 20352707950277

Offset: 0

Alois P. Heinz, Jan 22 2017

nonn

			a(0) = 1: [], the empty partition.
a(1) = 1: [1].
a(2) = 1: [1,3].
a(3) = 2: [1,3,5], [9].
a(4) = 5: [1,3,5,7], [7,9], [5,11], [3,13], [1,15].
a(5) = 12: [1,3,5,7,9], [5,9,11], [5,7,13], [3,9,13], [1,11,13], [3,7,15], [1,9,15], [3,5,17], [1,7,17], [1,5,19], [1,3,21], [25].

Chai Wah Wu, Table of n, a(n) for n = 0..507 (terms 0..200 from Alois P. Heinz)

Cf. A000009, A000290, A000700, A005408, A072243, A107379.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(d*
      [0, 1, -1, 1][1+irem(d, 4)], d=divisors(j)), j=1..n)/n)
    end:
a:= n-> b(n^2):
seq(a(n), n=0..30);

b[n_] := b[n] = If[n==0, 1, Sum[b[n-j]*Sum[d*{0, 1, -1, 1}[[1+Mod[d, 4]]], {d, Divisors[j]}], {j, 1, n}]/n];
a[n_] := b[n^2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

a(n) = [x^(n^2)] Product_{j>=0} (1 + x^(2*j+1)).
a(n) = A000700(A000290(n)).
a(n) ~ exp(Pi*n/sqrt(6)) / (2^(7/4) * 3^(1/4) * n^(3/2)). - Vaclav Kotesovec, Apr 10 2017

A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 68, 144, 312, 686, 1523, 3405, 7652, 17284, 39246, 89552, 205253, 472297, 1090544, 2525904, 5867037, 13663248, 31896309, 74628130, 174972341, 411032475, 967307190, 2280248312, 5383723722, 12729879673, 30141755384, 71462883813

Offset: 0

Moshe Shmuel Newman, Feb 15 2007

nonn

Also the number of self-conjugate partitions of the n-th triangular number.

			The 5th triangular number is 15. Writing this as a sum of distinct odd numbers: 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3 are all the possibilities. So a(5) = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Sequences A066655 and A104383 do the same thing for triangular numbers, with partitions or distinct partitions. Sequences A072213 and A072243 are analogs for squares rather than triangular numbers.

Cf. A000217.

g:= mul(1+x^(2*j+1),j=0..900): seq(coeff(g,x,n*(n+1)/2),n=0..40); # Emeric Deutsch, Feb 27 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i^2n, 0, b(n-2*i+1, i-1))))
    end:
a:= n-> b(n*(n+1)/2, ceil(n*(n+1)/4)*2-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 31 2018

a[n_] := SeriesCoefficient[QPochhammer[-x, x^2], {x, 0, n*(n+1)/2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 25 2018 *)

More terms from Emeric Deutsch, Feb 27 2007

a(0)=1 prepended by Alois P. Heinz, Jan 31 2018

A347621 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 distinct parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 8, 2, 1, 1, 1, 32, 192, 32, 3, 1, 1, 1, 390, 84756, 16444, 142, 4, 1, 1, 1, 16444, 5807301632, 11784471548, 3207086, 668, 5, 1, 1, 1, 4013544, 2496696209705056142, 16816734263788624008200, 74443865946867656, 1258238720, 3264, 6, 1

Offset: 0

Seiichi Manyama, Sep 09 2021

			Square array begins:
  1, 1,  1,     1,           1, ...
  1, 1,  1,     1,           1, ...
  1, 1,  2,     6,          32, ...
  1, 2,  8,   192,       84756, ...
  1, 2, 32, 16444, 11784471548, ...

Columns k=0..3 give A000012, A000009, A072243, A281501.
Rows n=0+1, 2-3 give A000012, A067735, A070235.
Main diagonal gives A064682.
Cf. A347615, A347630.

Table[If[n == k == 0, 1, PartitionsQ[#^k] &[n - k]], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Michael De Vlieger, Sep 09 2021 *)

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

T(n,k) = A000009(n^k).

A173493 Number of distinct squares that can be partitioned into distinct divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 1, 3, 3, 2, 1, 5, 1, 3, 4, 5, 1, 6, 1, 6, 3, 3, 1, 7, 2, 2, 4, 7, 1, 8, 1, 7, 3, 2, 2, 9, 1, 1, 3, 9, 1, 9, 1, 7, 7, 3, 1, 11, 2, 5, 2, 4, 1, 10, 2, 10, 2, 1, 1, 12, 1, 2, 7, 11, 1, 12, 1, 4, 2, 11, 1, 13, 1, 1, 9, 7, 1, 12, 1, 13, 6, 1, 1, 14, 1, 1, 2, 13, 1, 15, 1, 6, 2, 3, 3, 15, 1, 8

Offset: 1

Reinhard Zumkeller, Feb 20 2010

nonn

The partitions of the squares are generally not unique, see examples.

			divisors(9) = {1, 3, 9}: a(9) = #{1, 3+1, 9} = 3.
divisors(10) = {1, 2, 5, 10}: a(10) = #{1, 10+5+1} = 2.
divisors(12) = {1,2,3,4,6,12}: a(12) = #{1,4,9,16,25} = 5:
  2^2 = 4 = 3 + 1,
  3^2 = 6 + 3 = 6 + 2 + 1 = 4 + 3 + 2,
  4^2 = 12 + 4 = 12 + 3 + 1 = 6 + 4 + 3 + 2 + 1,
  5^2 = 12 + 6 + 4 + 3 = 12 + 6 + 4 + 2 + 1.
divisors(42)={1,2,3,6,7,14,21,42}: a(42)=#{k^2: 1<=k<=9}=9:
  2^2 = 3+1,
  3^2 = 7+2 = 6+3 = 6+2+1,
  4^2 = 14+2 = 7+6+3 = 7+6+2+1,
  5^2 = 21 + 3 + 1 = 14 + 7 + 3 + 1 = 14 + 6 + 3 + 2,
  6^2 = 21 + 14 + 1 = 21 + 7 + 6 + 2,
  7^2 = 42 + 7 = 42 + 6 + 1 = 21 + 14 + 7 + 6 + 1,
  8^2 = 42 + 21 + 1 = 42 + 14 + 7 + 1 = 42 + 14 + 6 + 2,
  9^2 = 42 + 21 + 14 + 3 + 1 = 42 + 21 + 7 + 6 + 3 + 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..250 from Reinhard Zumkeller)

Cf. A000203, A006532, A033630, A072243, A078705, A173494.

a[n_] := Module[{d = Divisors[n], sum, sq, x}, sum = Plus @@ d; sq = Range[Floor[Sqrt[sum]]]^2; Count[CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sq]], ?(# > 0&)]]; Array[a, 100] (* _Amiram Eldar, Apr 16 2025 *)

a(n) <= A078705(n).

a(A173494(n)) = 1.

A281501 Number of partitions of n^3 into distinct parts.

Original entry on oeis.org

1, 1, 6, 192, 16444, 3207086, 1258238720, 916112394270, 1168225267521350, 2496696209705056142, 8635565795744155161506, 46977052491046305327286932, 392416122247953159916295467008, 4931628582570689013431218105121792, 91603865924570978521516549662581412000

Offset: 0

Ilya Gutkovskiy, Jan 23 2017

nonn

			a(2) = 6 because we have [8], [7, 1], [6, 2], [5, 3], [5, 2, 1] and [4, 3, 1].

Vaclav Kotesovec, Table of n, a(n) for n = 0..117
Index entries for related partition-counting sequences

Cf. A000009, A000578, A072213, A072243, A128854.

Mathematica
```
Table[PartitionsQ[n^3], {n, 0, 10}]
```

a(n) = [x^(n^3)] Product_{k>=1} (1 + x^k).
a(n) = A000009(A000578(n)).
a(n) ~ exp(Pi*n^(3/2)/sqrt(3))/(4*3^(1/4)*n^(9/4)).

cp's OEIS Frontend

A072213 Number of partitions of n^2.

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A107379 Number of ways to write n^2 as the sum of n odd numbers, disregarding order.

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A281489 Number of partitions of n^2 into distinct odd parts.

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A126683 Number of partitions of the n-th triangular number n(n+1)/2 into distinct odd parts.

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Maple

Mathematica

Extensions

A347621 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 distinct parts.

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Mathematica

PARI

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A173493 Number of distinct squares that can be partitioned into distinct divisors of n.

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Mathematica

Formula

A281501 Number of partitions of n^3 into distinct parts.

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