|
%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
|
