A327484 -id:A327484 - cp's OEIS frontend

A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 1, 1, 7, 1, 1, 7, 5, 1, 1, 1, 1, 11, 15, 12, 6, 1, 1, 1, 1, 15, 7, 1, 1, 30, 19, 1, 1, 22, 34, 8, 1, 1, 1, 1, 30, 58, 27, 9, 1, 1, 1, 1, 42, 84, 64, 10, 1, 1, 105, 37, 1, 1, 56, 11, 1

Offset: 1

Gus Wiseman, Sep 13 2019

			Triangle begins:
  1
  1  1
  1  1
  1  2  1
  1  1
  1  3  3  1
  1  1
  1  5  4  1
  1  7  1
  1  7  5  1
  1  1
  1 11 15 12  6  1
  1  1
  1 15  7  1
  1 30 19  1
  1 22 34  8  1

Row sums are A067538.
The version for subsets is A327481.
Cf. A027750, A066571, A237984, A240850, A327474, A327483, A327484.

Table[Length[Select[IntegerPartitions[n],Mean[#]==d&]],{n,20},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 3, 7, 3, 1, 1, 3, 9, 9, 3, 1, 1, 3, 9, 19, 9, 3, 1, 1, 3, 9, 25, 25, 9, 3, 1, 1, 3, 9, 29, 51, 29, 9, 3, 1, 1, 3, 9, 31, 75, 75, 31, 9, 3, 1, 1, 3, 9, 31, 93, 151, 93, 31, 9, 3, 1, 1, 3, 9, 31, 105, 235, 235, 105, 31, 9, 3, 1

Offset: 1

Gus Wiseman, Sep 13 2019

All terms are odd.

			Triangle begins:
                         1
                       1   1
                     1   3   1
                   1   3   3   1
                 1   3   7   3   1
               1   3   9   9   3   1
             1   3   9  19   9   3   1
           1   3   9  25  25   9   3   1
         1   3   9  29  51  29   9   3   1
       1   3   9  31  75  75  31   9   3   1
     1   3   9  31  93 151  93  31   9   3   1
   1   3   9  31 105 235 235 105  31   9   3   1
The subsets counted in row n = 5:
  {1}  {2}      {3}          {4}      {5}
       {1,3}    {1,5}        {3,5}
       {1,2,3}  {2,4}        {3,4,5}
                {1,3,5}
                {2,3,4}
                {1,2,4,5}
                {1,2,3,4,5}

Row sums are A051293.
The sequence of rows converges to A066571.
The version for partitions is A327482.
Cf. A000016, A063776, A065795, A135342, A327474, A327475, A327483, A327484.

Table[Length[Select[Subsets[Range[n]],Mean[#]==k&]],{n,10},{k,n}]

A327483 Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 22, 34, 8, 1, 1, 231, 919, 249, 16, 1, 1, 8349, 112540, 55974, 1906, 32, 1, 1, 1741630, 107608848, 161410965, 4602893, 14905, 64, 1, 1, 4351078600, 1949696350591, 12623411092535, 676491536028, 461346215, 117874, 128, 1

Offset: 0

Gus Wiseman, Sep 13 2019

T(n,k) is the number of partitions of 2^n into 2^(n-k) parts. - Chai Wah Wu, Sep 21 2023

			Triangle begins:
      1
      1       1
      1       2         1
      1       5         4         1
      1      22        34         8       1
      1     231       919       249      16     1
      1    8349    112540     55974    1906    32  1
      1 1741630 107608848 161410965 4602893 14905 64 1
      ...

Alois P. Heinz, Rows n = 0..13, flattened

Row sums are A327484.
Column k = 1 is A068413 (shifted once to the right).
Cf. A067538, A237984, A240850, A327481, A327482.

Table[Length[Select[IntegerPartitions[2^n],Mean[#]==2^k&]],{n,0,5},{k,0,n}]

from sympy.utilities.iterables import partitions
from sympy import npartitions
def A327483_T(n,k):
    if k == 0 or k == n: return 1
    if k == n-1: return 1<Chai Wah Wu, Sep 21 2023

# uses A008284_T
def A327483_T(n,k): return A008284_T(1<Chai Wah Wu, Sep 21 2023

T(n+1,n) = 2^n for n >= 0. - Chai Wah Wu, Sep 14 2019

a(28)-a(35) from Chai Wah Wu, Sep 14 2019

Row n=8 from Alois P. Heinz, Sep 21 2023

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A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

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A327481 Triangle read by rows where T(n,k) is the number of nonempty subsets of {1..n} with mean k.

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A327483 Triangle read by rows where T(n,k) is the number of integer partitions of 2^n with mean 2^k, 0 <= k <= n.

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