A281489 -id:A281489 - cp's OEIS frontend

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

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

A072243 Number of distinct partitions of n^2.

Original entry on oeis.org

1, 1, 2, 8, 32, 142, 668, 3264, 16444, 84756, 444793, 2368800, 12769602, 69545358, 382075868, 2114965120, 11784471548, 66043042088, 372022512608, 2105220502772, 11962163400706, 68223286792200, 390406746862530, 2240962117491470, 12899456450932840

Offset: 0

Robert G. Wilson v, Jul 06 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1260 (terms 0..200 from Alois P. Heinz)

Cf. A000009, A000041, A000290, A072213, A281489.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d::odd, d, 0), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= n-> b(n^2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 22 2017

Mathematica
```
Table[ PartitionsQ[n^2], {n, 1, 24}]
```

a(n) ~ exp(Pi*n/sqrt(3)) / (4*3^(1/4)*n^(3/2)). - Vaclav Kotesovec, Dec 01 2015

a(n) = A000009(A000290(n)). - Alois P. Heinz, Jan 22 2017

a(0)=1 prepended by Alois P. Heinz, Jan 22 2017

A347626 Number of partitions of n^n into distinct odd parts.

Original entry on oeis.org

1, 1, 1, 14, 2824974, 7375247711025022789604527681

Offset: 0

Seiichi Manyama, Sep 09 2021

nonn

The next term a(6) = 1.46058224...*10^116 is too large to include.

Seiichi Manyama, Table of n, a(n) for n = 0..7

Main diagonal of A347630.

Cf. A000312, A000700, A064682, A281489.

a(n) = polcoef(prod(k=0, n^n\2, 1+x^(2*k+1)+x*O(x^(n^n))), n^n);

a(n) = [x^(n^n)] Product_{k>=0} (1 + x^(2*k+1)).

a(n) = A000700(n^n).

A347630 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 odd parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 5, 14, 5, 1, 1, 1, 1, 23, 833, 276, 12, 1, 1, 1, 1, 276, 1731778, 2824974, 9912, 33, 1, 1, 1, 1, 11564, 1741020966255, 824068326214949, 150145281903, 602245, 93, 2, 1, 1, 1, 2824974, 78444810948209793568790, 195321031346209256918890884699755, 7375247711025022789604527681, 116880108216597935, 57638873, 276, 2, 1

Offset: 0

Seiichi Manyama, Sep 09 2021

			Square array begins:
  1, 1,  1,    1,            1,                            1, ...
  1, 1,  1,    1,            1,                            1, ...
  1, 0,  1,    2,            5,                           23, ...
  1, 1,  2,   14,          833,                      1731778, ...
  1, 1,  5,  276,      2824974,              824068326214949, ...
  1, 1, 12, 9912, 150145281903, 7375247711025022789604527681, ...

Columns k=0..2 give A000012, A000700, A281489.
Main diagonal gives A347626.
Cf. A347621.

T(n, k) = polcoef(prod(j=0, n^k\2, 1+x^(2*j+1)+x*O(x^(n^k))), n^k);

T(n,k) = A000700(n^k).

cp's OEIS Frontend

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

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A072243 Number of distinct partitions of n^2.

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Maple

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A347626 Number of partitions of n^n into distinct odd parts.

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A347630 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 odd parts.

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