A180653 'DP(n,k)' triangle read by rows. DP(n,k) is the number of k-double-palindromes of n.

0, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 4, 4, 1, 0, 5, 3, 8, 4, 1, 0, 6, 6, 12, 12, 6, 1, 0, 7, 6, 17, 12, 19, 6, 1, 0, 8, 7, 24, 24, 20, 24, 8, 1, 0, 9, 8, 32, 21, 50, 24, 32, 8, 1, 0, 10, 10, 40, 40, 60, 60, 40, 40, 10, 1, 0, 11, 9, 49, 40, 100, 60, 98, 35, 51, 10, 1

Offset

1,5

Comments

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. A palindrome is a word which is the same when written backwards.

A k-double-palindrome of n is a k-composition of n which is the concatenation of two palindromes, PP'=P|P', where both |P|, |P'|>=1.

For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes.

Let DP(n,k) denote the number of k-double-palindromes of n.

This sequence is the 'DP(n,k)' triangle read by rows.

References

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

Links

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

Formula

T(n,k) = A180279(n,k) - A179519(n,k). - Andrew Howroyd, Sep 27 2019

Example

The triangle begins
  0
  0 1
  0 2 1
  0 3 2  1
  0 4 4  4  1
  0 5 3  8  4  1
  0 6 6 12 12  6  1
  0 7 6 17 12 19  6  1
  0 8 7 24 24 20 24  8 1
  0 9 8 32 21 50 24 32 8 1
  ...
For example, row 8 is: 0 7 6 17 12 19 6 1.
We have DP(8,3)=6 because there are 6 3-double-palindromes of 8: 116, 611, 224, 422, 233, and 332.
We have DP(8,4)=17 because there are 17 4-double-palindromes of 8: 1115, 5111, 1511, 1151, 1214, 4121, 1412, 2141, 1133, 3311, 1313, 3131, 1232, 2123, 3212, 2321, and 2222.

Prog

(PARI) \\ p(n, k) is k*A119963(n, k); q(n, k) is A051159(n-1, k-1).
p(n, k) = {k*binomial((n-k%2)\2, k\2)}
q(n, k) = {if(n%2==1&&k%2==0, 0, binomial((n-1)\2, (k-1)\2))}
invphi(n) = {sumdiv(n, d, d*moebius(d))}
T(n, k) = sumdiv(gcd(n, k), d, invphi(d) * p(n/d, k/d) - moebius(d) * q(n/d, k/d)); \\ Andrew Howroyd, Sep 27 2019

Crossrefs

Row sums are A180750.

See sequence A051159 for the triangle whose (n, k) term gives the number of k-palindromes (single-palindromes) of n.

Cf. A179519, A180279, A180918, A181111, A181169.

Sequence in context: A025647 A025653 A131103 * A259100 A368091 A365004

Adjacent sequences: A180650 A180651 A180652 * A180654 A180655 A180656

Keyword

nonn,tabl

Author

John P. McSorley, Sep 14 2010

Extensions

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

Status

approved