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.

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

0,5

Comments

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

Links

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

Formula

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

Example

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
      ...

Mathematica

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

Prog

(Python)
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<<n-1
    if k == 1: return npartitions(1<<n-1)
    a, b = 1<<n, 1<<n-k
    return sum(1 for s, p in partitions(a, m=b, size=True) if s==b) # Chai Wah Wu, Sep 21 2023
(Python)
# uses A008284_T
def A327483_T(n, k): return A008284_T(1<<n, 1<<n-k) # Chai Wah Wu, Sep 21 2023

Crossrefs

Row sums are A327484.

Column k = 1 is A068413 (shifted once to the right).

Cf. A067538, A237984, A240850, A327481, A327482.

Sequence in context: A220738 A284732 A371766 * A327884 A050145 A222573

Adjacent sequences: A327480 A327481 A327482 * A327484 A327485 A327486

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 13 2019

Extensions

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

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

Status

approved