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A327483
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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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5
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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
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OFFSET
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0,5
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COMMENTS
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T(n,k) is the number of partitions of 2^n into 2^(n-k) parts. - Chai Wah Wu, Sep 21 2023
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LINKS
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FORMULA
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EXAMPLE
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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
...
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[2^n], Mean[#]==2^k&]], {n, 0, 5}, {k, 0, n}]
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PROG
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(Python)
from sympy.utilities.iterables import partitions
from sympy import npartitions
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)
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CROSSREFS
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Column k = 1 is A068413 (shifted once to the right).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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