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%I #9 May 23 2021 00:20:08
%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,4,1,1,
%T 1,5,1,1,1,2,1,1,1,2,2,1,1,4,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,7,1,1,1,2,
%U 1,1,1,5,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N Number of palindromic factorizations of n.
%C A palindrome is a sequence that is the same whether it is read forward or in reverse. A palindromic factorization of n is a finite multiset of positive integers > 1 with product n that can be permuted into a palindrome.
%F a(2^n) = A025065(n).
%F a(n) = A057567(A000188(n)). - _Andrew Howroyd_, May 22 2021
%e The palindromic factorizations for n = 2, 4, 16, 36, 64, 144:
%e (2) (4) (16) (36) (64) (144)
%e (2*2) (4*4) (6*6) (8*8) (12*12)
%e (2*2*4) (2*2*9) (4*4*4) (4*4*9)
%e (2*2*2*2) (3*3*4) (2*2*16) (4*6*6)
%e (2*2*3*3) (2*2*4*4) (2*2*36)
%e (2*2*2*2*4) (3*3*16)
%e (2*2*2*2*2*2) (2*2*6*6)
%e (3*3*4*4)
%e (2*2*2*2*9)
%e (2*2*3*3*4)
%e (2*2*2*2*3*3)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t palQ[y_]:=Select[Permutations[y],#==Reverse[#]&]!={};
%t Table[Length[Select[facs[n],palQ]],{n,50}]
%Y Positions of 1's are A005117.
%Y The case of palindromic compositions is A016116.
%Y The additive version (palindromic partitions) is A025065.
%Y The case of palindromic prime signature is A242414.
%Y The case of palindromic plane trees is A319436.
%Y A001055 counts factorizations.
%Y A229153 ranks non-palindromic partitions.
%Y A265640 ranks palindromic partitions.
%Y Cf. A000041, A000070, A000188, A004526, A030229, A056503, A057567, A082293, A344414.
%K nonn
%O 1,4
%A _Gus Wiseman_, May 22 2021