

A179519


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


6



1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 1, 0, 1, 0, 3, 0, 3, 0, 0, 1, 0, 3, 2, 3, 2, 1, 0, 1, 0, 3, 0, 6, 0, 4, 0, 0, 1, 0, 4, 4, 5, 4, 4, 4, 1, 0, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 0, 1, 0, 4, 4, 10, 8, 10, 8, 4, 4, 1, 0
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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.
A kcomposition is aperiodic (primitive) if its period is k, or if it is not the concatenation of a smaller composition.
A kpalindrome of n is a kcomposition of n which is a palindrome.
Let AP(n,k) denote the number of aperiodic kpalindromes of n.
This sequence is the 'AP(n,k)' triangle read by rows.
The g.f. of this triangular array follows easily from A. Howroyd's formula for this sequence and P. Deleham's g.f. for sequence A051159. If T(n,k) = A051159(n,k), then g.f. = Sum_{n,k>=1} AP(n,k)*x^n*y^k = Sum_{n,k>=1} Sum_{dgcd(n,k)} mu(d)*T(n/d1,k/d1)*x^n*y^k. Letting m = n/d and s = k/d, we get g.f. = Sum_{d>=1} mu(d)*Sum_{m,s>=1} T(m1,s1)*(x^d)^m*(y^d)^s. But P. Deleham's formula for sequence A051159 implies Sum_{m,s>=1} T(m1,s1)*x^m*y^s = x*y*(1+x+x*y)/(1x^2x^2*y^2). Thus, Sum_{n,k>=1} AP(n,k)*x^n*y^k = Sum_{d>=1} mu(d)*f(x^d,y^d), where f(x,y) = x*y*(1+x+x*y)/(1x^2x^2*y^2).  Petros Hadjicostas, Nov 04 2017


REFERENCES

John P. McSorley: Counting kcompositions of n with palindromic and related structures. Preprint, 2010.


LINKS



FORMULA

G.f.: Sum_{n>=1} mu(n)*f(x^n,y^n), where f(x,y) = x*y*(1+x+x*y)/(1x^2x^2*y^2).  Petros Hadjicostas, Nov 04 2017


EXAMPLE

The triangle begins
1
1,0
1,0,0
1,0,1,0
1,0,2,0,0
1,0,1,2,1,0
1,0,3,0,3,0,0
1,0,3,2,3,2,1,0
1,0,3,0,6,0,4,0,0
1,0,4,4,5,4,4,4,1,0
For example, row 8 is 1,0,3,2,3,2,1,0.
We have AP(8,3)=3 because there are 3 aperiodic 3palindromes of 8, namely: 161, 242, and 323.
We have AP(8,4)=2 because there are 2 aperiodic 4palindromes of 8, namely: 3113 and 1331.


MATHEMATICA

T[n_, k_] := Sum[MoebiusMu[d]*QBinomial[n/d1, k/d1, 1], {d, Divisors[ GCD[n, k]]}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* JeanFrançois Alcover, Oct 30 2017, after Andrew Howroyd *)


PROG

(PARI) \\ here p(n, k)=A051159(n1, k1) is number of kpalindromes of n.
p(n, k) = if(n%2==1&&k%2==0, 0, binomial((n1)\2, (k1)\2));
T(n, k) = sumdiv(gcd(n, k), d, moebius(d) * p(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 count the aperiodic kpalindromes of n up to cyclic equivalence, APE(n, k), we get sequence A179317.


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STATUS

approved



