

A181111


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


5



0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 4, 4, 0, 0, 4, 2, 6, 4, 0, 0, 6, 6, 12, 12, 6, 0, 0, 6, 6, 14, 12, 16, 6, 0, 0, 8, 6, 24, 24, 18, 24, 8, 0, 0, 8, 8, 28, 20, 44, 24, 28, 8, 0, 0, 10, 10, 40, 40, 60, 60, 40, 40, 10, 0, 0, 10, 8, 44, 40, 94, 60, 88, 32, 46, 10, 0
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OFFSET

1,5


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, i.e., if it is not the concatenation of at least two smaller compositions.
A palindrome is a word which is the same when written backwards.
A kdoublepalindrome of n is a kcomposition of n which is the concatenation of two palindromes, PP'=PP', where both P, P'>=1.
For example 1123532=1123532 is a 7doublepalindrome of 17 since both 11 and 23532 are palindromes. It is also aperiodic, and so it is an aperiodic 7doublepalindrome of 17. The 4doublepalindrome of 8 1313=1313 is not aperiodic, so it is not an aperiodic 4doublepalindrome of 8.
Let ADP(n,k) denote the number of aperiodic kdoublepalindromes of n.
This sequence is the 'ADP(n,k)' triangle read by rows.


REFERENCES

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


LINKS



FORMULA



EXAMPLE

The triangle begins:
0
0 0
0 2 0
0 2 2 0
0 4 4 4 0
0 4 2 6 4 0
0 6 6 12 12 6 0
0 6 6 14 12 16 6 0
0 8 6 24 24 18 24 8 0
0 8 8 28 20 44 24 28 8 0
...
For example, row 8 is: 0 6 6 14 12 16 6 0.
We have ADP(8,3)=6 because there are 6 aperiodic 3doublepalindromes of 8: 116, 611, 224, 422, 233, and 332.
We have ADP(8,4)=14 because there are 14 4doublepalindromes of 8: 1115, 5111, 1511, 1151, 1214, 4121, 1412, 2141, 1133, 3311, 1232, 2123, 3212, and 2321.


PROG

(PARI) \\ here p(n, k) is A119963(n, k), q(n, k) is A051159(n1, k1).
p(n, k) = { binomial((nk%2)\2, k\2) }
q(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) * (k*p(n/d, k/d)  q(n/d, k/d))); \\ Andrew Howroyd, Sep 27 2019


CROSSREFS

See sequence A180653 for the triangle whose (n, k) term gives the number of kdoublepalindromes of n.
See sequence A179519 for the triangle whose (n, k) term gives the number of aperiodic kpalindromes (singlepalindromes) of n.


KEYWORD



AUTHOR



EXTENSIONS

a(37) corrected and terms a(56) and beyond from Andrew Howroyd, Sep 27 2019


STATUS

approved



