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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.

%I A327483 #31 Sep 21 2023 19:30:51
%S A327483 1,1,1,1,2,1,1,5,4,1,1,22,34,8,1,1,231,919,249,16,1,1,8349,112540,
%T A327483 55974,1906,32,1,1,1741630,107608848,161410965,4602893,14905,64,1,1,
%U A327483 4351078600,1949696350591,12623411092535,676491536028,461346215,117874,128,1
%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.
%C A327483 T(n,k) is the number of partitions of 2^n into 2^(n-k) parts. - _Chai Wah Wu_, Sep 21 2023
%H A327483 Alois P. Heinz, <a href="/A327483/b327483.txt">Rows n = 0..13, flattened</a>
%F A327483 T(n+1,n) = 2^n for n >= 0. - _Chai Wah Wu_, Sep 14 2019
%e A327483 Triangle begins:
%e A327483       1
%e A327483       1       1
%e A327483       1       2         1
%e A327483       1       5         4         1
%e A327483       1      22        34         8       1
%e A327483       1     231       919       249      16     1
%e A327483       1    8349    112540     55974    1906    32  1
%e A327483       1 1741630 107608848 161410965 4602893 14905 64 1
%e A327483       ...
%t A327483 Table[Length[Select[IntegerPartitions[2^n],Mean[#]==2^k&]],{n,0,5},{k,0,n}]
%o A327483 (Python)
%o A327483 from sympy.utilities.iterables import partitions
%o A327483 from sympy import npartitions
%o A327483 def A327483_T(n,k):
%o A327483     if k == 0 or k == n: return 1
%o A327483     if k == n-1: return 1<<n-1
%o A327483     if k == 1: return npartitions(1<<n-1)
%o A327483     a, b = 1<<n, 1<<n-k
%o A327483     return sum(1 for s,p in partitions(a,m=b,size=True) if s==b) # _Chai Wah Wu_, Sep 21 2023
%o A327483 (Python)
%o A327483 # uses A008284_T
%o A327483 def A327483_T(n,k): return A008284_T(1<<n,1<<n-k) # _Chai Wah Wu_, Sep 21 2023
%Y A327483 Row sums are A327484.
%Y A327483 Column k = 1 is A068413 (shifted once to the right).
%Y A327483 Cf. A067538, A237984, A240850, A327481, A327482.
%K A327483 nonn,tabl
%O A327483 0,5
%A A327483 _Gus Wiseman_, Sep 13 2019
%E A327483 a(28)-a(35) from _Chai Wah Wu_, Sep 14 2019
%E A327483 Row n=8 from _Alois P. Heinz_, Sep 21 2023