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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
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.
Sequence in context: A025647 A025653 A131103 * A259100 A368091 A365004
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.)