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 A179181 'PE(n,k)' triangle read by rows. PE(n,k) is the number of k-palindromes of n up to cyclic equivalence. 3
 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 2, 3, 2, 1, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 1, 4, 2, 6, 4, 4, 2, 1, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 1, 5, 3, 10, 6, 10, 5, 5, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,13 COMMENTS A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. Two k-compositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts. A k-palindrome of n is a k-composition of n which is a palindrome. Let PE(n,k) denote the number of k-palindromes of n up to cyclic equivalence. This sequence is the 'PE(n,k)' triangle read by rows. The triangle begins 1 1,1 1,0,1 1,1,1,1 1,0,2,0,1 1,1,2,1,1,1 1,0,3,0,3,0,1 1,1,3,2,3,2,1,1 1,0,4,0,6,0,4,0,1 1,1,4,2,6,4,4,2,1,1 For example, row 8 is 1,1,3,2,3,2,1,1. We have PE(8,3)=3 because there are 3 3-palindromes of 8, namely: 161, 242, and 323, and none are cyclically equivalent to the others. We have PE(8,4)=2 because there are 3 4-palindromes of 8, namely: 3113, 1331, and 2222, but 3113 and 1331 are cyclically equivalent. REFERENCES John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 FORMULA PE(n, k) = Sum_{d | gcd(n,k)} A179317(n/d, k/d). - Andrew Howroyd, Oct 07 2017 PROG (PARI) p(n, k) = if(n%2==1&&k%2==0, 0, binomial((n-1)\2, (k-1)\2)); APE(n, k) = if(k%2, 1, 1/2) * sumdiv(gcd(n, k), d, moebius(d) * p(n/d, k/d)); T(n, k) = sumdiv(gcd(n, k), d, APE(n/d, k/d)); for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Oct 07 2017 CROSSREFS If we ignore cyclic equivalence then we have sequence A051159. The row sums of the 'PE(n, k)' triangle give sequence A056503. Cf. A179317, A179519. Sequence in context: A258832 A121444 A118230 * A153246 A025889 A126306 Adjacent sequences:  A179178 A179179 A179180 * A179182 A179183 A179184 KEYWORD nonn,tabl AUTHOR John P. McSorley, Jun 30 2010 EXTENSIONS Terms a(56) and beyond from Andrew Howroyd, Oct 07 2017 STATUS approved

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Last modified May 30 15:38 EDT 2020. Contains 334726 sequences. (Running on oeis4.)