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 A180653 'DP(n,k)' triangle read by rows. DP(n,k) is the number of k-double-palindromes of n. 5
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified February 28 16:26 EST 2024. Contains 370400 sequences. (Running on oeis4.)