A327483 - OEIS

%I #31 Sep 21 2023 19:30:51

%S 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 55974,1906,32,1,1,1741630,107608848,161410965,4602893,14905,64,1,1,

%U 4351078600,1949696350591,12623411092535,676491536028,461346215,117874,128,1

%N 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 T(n,k) is the number of partitions of 2^n into 2^(n-k) parts. - _Chai Wah Wu_, Sep 21 2023

%H Alois P. Heinz, <a href="/A327483/b327483.txt">Rows n = 0..13, flattened</a>

%F T(n+1,n) = 2^n for n >= 0. - _Chai Wah Wu_, Sep 14 2019

%e Triangle begins:

%e       1

%e       1       1

%e       1       2         1

%e       1       5         4         1

%e       1      22        34         8       1

%e       1     231       919       249      16     1

%e       1    8349    112540     55974    1906    32  1

%e       1 1741630 107608848 161410965 4602893 14905 64 1

%e       ...

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

%o (Python)

%o from sympy.utilities.iterables import partitions

%o from sympy import npartitions

%o def A327483_T(n,k):

%o     if k == 0 or k == n: return 1

%o     if k == n-1: return 1<<n-1

%o     if k == 1: return npartitions(1<<n-1)

%o     a, b = 1<<n, 1<<n-k

%o     return sum(1 for s,p in partitions(a,m=b,size=True) if s==b) # _Chai Wah Wu_, Sep 21 2023

%o (Python)

%o # uses A008284_T

%o def A327483_T(n,k): return A008284_T(1<<n,1<<n-k) # _Chai Wah Wu_, Sep 21 2023

%Y Row sums are A327484.

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

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

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Sep 13 2019

%E a(28)-a(35) from _Chai Wah Wu_, Sep 14 2019

%E Row n=8 from _Alois P. Heinz_, Sep 21 2023