

A179181


'PE(n,k)' triangle read by rows. PE(n,k) is the number of kpalindromes 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
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OFFSET

1,13


COMMENTS

A kcomposition of n is an ordered collection of k positive integers (parts) which sum to n.
Two kcompositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.
A kpalindrome of n is a kcomposition of n which is a palindrome.
Let PE(n,k) denote the number of kpalindromes 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 3palindromes 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 4palindromes of 8, namely: 3113, 1331, and 2222, but 3113 and 1331 are cyclically equivalent.


REFERENCES

John P. McSorley: Counting kcompositions 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((n1)\2, (k1)\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



