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A375783 Number of partitions with weight fixed by reversal.

%I A375783 #12 Aug 30 2024 09:45:51
%S A375783 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,
%T A375783 120,2,76,90,18,198,320,2,20,142,388,2,478,2,478,872,24,2,956,506,652,
%U A375783 304,988,2,1852,1944,2210,418,30,2,5528,2,32,5168,2518,5032
%N A375783 Number of partitions with weight fixed by reversal.
%H A375783 Chai Wah Wu, <a href="/A375783/b375783.txt">Table of n, a(n) for n = 1..286</a>
%H A375783 David J. Hemmer and Karlee J. Westrem, <a href="https://arxiv.org/abs/2402.02250">Palindrome Partitions and the Calkin-Wilf Tree</a>, arXiv:2402.02250 [math.CO], 2024. See Table 5.1 p. 9.
%F A375783 a(n) = 2 if and only if n = 3 or n + 1 > 2 is prime (Hemmer and Westrem).
%F A375783 For proofs of the following, see A368548.
%F A375783 Let T(n,k) be the table in A183917.
%F A375783 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.
%F A375783 Let y = 2*Sum_{d|n+1, d>=3, and d is odd} T(d-2,(n+1)/d-1).
%F A375783 Then a(n) = x+y.
%F A375783 If n>3 is odd and (n+1)/2 is prime, then a(n) = A368548(n) = (n+3)/2.
%F A375783 a(2^n-1) = Sum_{i=0..n-1} T(2^(n-i)-2,2^i-1).
%o A375783 (Python)
%o A375783 from sympy import divisors
%o A375783 from sympy.utilities.iterables import partitions
%o A375783 def A375783(n):
%o A375783     def A183917_T(n,k): return sum(1 for p in partitions(k*n,m=n,k=k<<1))
%o A375783     x = sum(A183917_T((n+1)//d-2,d-1) for d in divisors(n+1>>1, generator=True)) if n&1 else 0
%o A375783     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
%o A375783     return x+y
%Y A375783 Cf. A183917, A368548.
%K A375783 nonn
%O A375783 1,2
%A A375783 _Chai Wah Wu_, Aug 28 2024