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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.

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.

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

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

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

...

Andrew Howroyd, <a href="/A180653/b180653.txt">Table of n, a(n) for n = 1..1275</a>

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

(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

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

Row sums are A180750.

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

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

For example , row 8 is: 0 7 6 17 12 19 6 1.

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

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