A375783 Number of partitions with weight fixed by reversal.
1, 2, 2, 2, 4, 2, 4, 4, 6, 2, 10, 2, 8, 14, 10, 2, 20, 2, 28, 28, 12, 2, 52, 28, 14, 52, 80, 2, 120, 2, 76, 90, 18, 198, 320, 2, 20, 142, 388, 2, 478, 2, 478, 872, 24, 2, 956, 506, 652, 304, 988, 2, 1852, 1944, 2210, 418, 30, 2, 5528, 2, 32, 5168, 2518, 5032
Offset: 1
Keywords
Links
- Chai Wah Wu, Table of n, a(n) for n = 1..286
- David J. Hemmer and Karlee J. Westrem, Palindrome Partitions and the Calkin-Wilf Tree, arXiv:2402.02250 [math.CO], 2024. See Table 5.1 p. 9.
Programs
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Python
from sympy import divisors from sympy.utilities.iterables import partitions def A375783(n): def A183917_T(n,k): return sum(1 for p in partitions(k*n,m=n,k=k<<1)) x = sum(A183917_T((n+1)//d-2,d-1) for d in divisors(n+1>>1, generator=True)) if n&1 else 0 y = sum(A183917_T(d-2,(n+1)//d-1) for d in divisors((n+1)>>(~(n+1)&n).bit_length(), generator=True) if d>=3)<<1 return x+y
Formula
a(n) = 2 if and only if n = 3 or n + 1 > 2 is prime (Hemmer and Westrem).
For proofs of the following, see A368548.
Let T(n,k) be the table in A183917.
Let x = 0 if n is even and x = Sum_{d|(n+1)/2} T((n+1)/d-2,d-1) if n is odd.
Let y = 2*Sum_{d|n+1, d>=3, and d is odd} T(d-2,(n+1)/d-1).
Then a(n) = x+y.
If n>3 is odd and (n+1)/2 is prime, then a(n) = A368548(n) = (n+3)/2.
a(2^n-1) = Sum_{i=0..n-1} T(2^(n-i)-2,2^i-1).