

A181169


'ADPE(n,k)' triangle read by rows. ADPE(n,k) is the number of aperiodic kdoublepalindromes of n up to cyclic equivalence.


4



0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 1, 0, 0, 2, 1, 2, 1, 0, 0, 3, 3, 3, 3, 1, 0, 0, 3, 3, 4, 3, 3, 1, 0, 0, 4, 3, 6, 6, 3, 4, 1, 0, 0, 4, 4, 8, 5, 8, 4, 4, 1, 0, 0, 5, 5, 10, 10, 10, 10, 5, 5, 1, 0, 0, 5, 4, 12, 10, 17, 10, 12, 4, 5, 1, 0, 0, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1, 0
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OFFSET

1,12


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. See sequence A181111.
Two kcompositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.
Let ADPE(n,k) denote the number of aperiodic kdoublepalindromes of n up to cyclic equivalence, i.e., the number of cyclic equivalence classes containing at least one aperiodic kdoublepalindrome.
This sequence is the `ADPE(n,k)' triangle read by rows.
The triangle begins:
0
0 0
0 1 0
0 1 1 0
0 2 2 1 0
0 2 1 2 1 0
0 3 3 3 3 1 0
0 3 3 4 3 3 1 0
0 4 3 6 6 3 4 1 0
0 4 4 8 5 8 4 4 1 0
...
For example, row 8 is: 0 3 3 4 3 3 1 0.
We have ADPE(8,3)=3 because the 6 aperiodic 3doublepalindromes of 8: 116, 611, 224, 422, 233, and 332 come in 3 cyclic equivalence classes: {116, 611, 161}, {224, 422, 242}, and {233, 323, 332}.
We have ADPE(8,4)=4 because there are 4 4doublepalindromes of 8 up to cyclic equivalence, the 4 classes are: {1115, 5111, 1511, 1151}, {1214, 4121, 1412, 2141}, {1133, 3113, 3311, 1331}, and {1232, 2123, 3212, 2321}.


REFERENCES

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


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275


FORMULA

T(n, 1) = 0; T(n, k) = A180424(n, k) for k > 1.  Andrew Howroyd, Sep 28 2019


PROG

(PARI) \\ here RE(n, k) is A119963(n, k).
RE(n, k) = binomial((nk%2)\2, k\2);
T(n, k) = if(k<=1, 0, sumdiv(gcd(n, k), d, moebius(d)*RE(n/d, k/d))); \\ Andrew Howroyd, Sep 28 2019


CROSSREFS

Row sums are A181314.
If we remove the aperiodic requirement we get sequence A180918.
If we count the aperiodic kdoublepalindromes of n (not the number of classes) we get sequence A181111 which is the 'ADP(n, k)' triangle read by rows, where ADP(n, k) is the number of aperiodic kdoublepalindromes of n.
Cf. A180424.
Sequence in context: A287385 A191411 A133418 * A029390 A108040 A137566
Adjacent sequences: A181166 A181167 A181168 * A181170 A181171 A181172


KEYWORD

nonn,tabl


AUTHOR

John P. McSorley, Oct 07 2010


EXTENSIONS

Terms a(56) and beyond from Andrew Howroyd, Sep 27 2019


STATUS

approved



