A180653 -id:A180653 - cp's OEIS frontend

A180918 'DPE(n,k)' triangle read by rows. DPE(n,k) is the number of k-double-palindromes of n up to cyclic equivalence.

0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 1, 0, 3, 2, 3, 1, 1, 0, 3, 3, 3, 3, 1, 1, 0, 4, 3, 6, 3, 4, 1, 1, 0, 4, 4, 6, 6, 4, 4, 1, 1, 0, 5, 4, 10, 6, 10, 4, 5, 1, 1, 0, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 0, 6, 5, 15, 10, 20, 10, 15, 5, 6, 1, 1, 0, 6, 6, 15, 15, 20, 20, 15, 15, 6, 6, 1, 1

Offset: 1

John P. McSorley, Sep 23 2010

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. Two k-compositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.
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.
See sequence A180653. For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes.
Let DPE(n,k) denote the number of k-double-palindromes of n up to cyclic equivalence.
This sequence is the 'DPE(n,k)' triangle read by rows.

			The triangle begins:
  0
  0 1
  0 1 1
  0 2 1 1
  0 2 2 1 1
  0 3 2 3 1 1
  0 3 3 3 3 1 1
  0 4 3 6 3 4 1 1
  0 4 4 6 6 4 4 1 1
  0 5 4 10 6 10 4 5 1 1
  ...
For example, row 8 is: 0 4 3 6 3 4 1 1.
We have DPE(8,3)=3 because there are 3 3-double-palindromes of 8 up to cyclic equivalence: {116, 611}, {224, 422}, and {233, 332}.
We have DPE(8,4)=6 because there are 6 4-double-palindromes of 8: up to cyclic equivalence: {1115, 5111, 1511, 1151}, {1214, 4121, 1412, 2141}, {1133, 3311}, {1313, 3131}, {1232, 2123, 3212, 2321}, and {2222}.

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

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

Row sums are A027383(n-1).
If we remove the cyclic equivalence requirement, and just count k-double-palindromes of n, then we get sequence A180653.
If we replace the left hand column of 0's by 1's in the triangle above, we get the triangle 'RE(n, k)' where RE(n, k) is the number of k-reverses of n up to cyclic equivalence, see the McSorley reference above for more details and also sequence A119963.
See sequence A179181 for the triangle whose (n, k) term gives the number of k-palindromes (single-palindromes) of n up to cyclic equivalence.

T(n, k) = {if(k<=1, 0, binomial((n-k%2)\2, k\2))} \\ Andrew Howroyd, Sep 27 2019

T(n, 1) = 0; T(n, k) = A119963(n,k) for k > 1.

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

A181111 'ADP(n,k)' triangle read by rows. ADP(n,k) is the number of aperiodic k-double-palindromes of n.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 4, 4, 0, 0, 4, 2, 6, 4, 0, 0, 6, 6, 12, 12, 6, 0, 0, 6, 6, 14, 12, 16, 6, 0, 0, 8, 6, 24, 24, 18, 24, 8, 0, 0, 8, 8, 28, 20, 44, 24, 28, 8, 0, 0, 10, 10, 40, 40, 60, 60, 40, 40, 10, 0, 0, 10, 8, 44, 40, 94, 60, 88, 32, 46, 10, 0

Offset: 1

John P. McSorley, Oct 03 2010

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. A k-composition 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 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. It is also aperiodic, and so it is an aperiodic 7-double-palindrome of 17. The 4-double-palindrome of 8 1313=131|3 is not aperiodic, so it is not an aperiodic 4-double-palindrome of 8.
Let ADP(n,k) denote the number of aperiodic k-double-palindromes of n.
This sequence is the 'ADP(n,k)' triangle read by rows.

			The triangle begins:
  0
  0 0
  0 2 0
  0 2 2  0
  0 4 4  4  0
  0 4 2  6  4  0
  0 6 6 12 12  6  0
  0 6 6 14 12 16  6  0
  0 8 6 24 24 18 24  8 0
  0 8 8 28 20 44 24 28 8 0
  ...
For example, row 8 is: 0 6 6 14 12 16 6 0.
We have ADP(8,3)=6 because there are 6 aperiodic 3-double-palindromes of 8: 116, 611, 224, 422, 233, and 332.
We have ADP(8,4)=14 because there are 14 4-double-palindromes of 8: 1115, 5111, 1511, 1151, 1214, 4121, 1412, 2141, 1133, 3311, 1232, 2123, 3212, and 2321.

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

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

Row sums are A181135.
See sequence A180653 for the triangle whose (n, k) term gives the number of k-double-palindromes of n.
See sequence A179519 for the triangle whose (n, k) term gives the number of aperiodic k-palindromes (single-palindromes) of n.
Cf. A180279, A180918, A181169.

\\ here p(n,k) is A119963(n,k), q(n,k) is A051159(n-1, k-1).
p(n, 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)) }
T(n, k) = sumdiv(gcd(n, k), d, moebius(d) * (k*p(n/d, k/d) - q(n/d, k/d))); \\ Andrew Howroyd, Sep 27 2019

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

a(37) corrected and terms a(56) and beyond from Andrew Howroyd, Sep 27 2019

A294332 G.f.: exp( Sum_{n>=1} A180563(n) * x^n / n ).

Original entry on oeis.org

1, 1, -1, 5, -45, 609, -11141, 257281, -7170355, 233936995, -8744103079, 368479396171, -17288353555771, 894005702731735, -50527305282004435, 3099060459670425655, -205028564671300495120, 14554510561318327509610, -1103542106915790217739110, 89009707681627448130203830, -7610129271299704960998906454, 687495658528174987634449288846, -65438091790081511530153327883206, 6545685493719560524729653911676430

Offset: 0

Paul D. Hanna, Oct 28 2017

sign

			G.f.: A(x) = 1 + x - x^2 + 5*x^3 - 45*x^4 + 609*x^5 - 11141*x^6 + 257281*x^7 - 7170355*x^8 + 233936995*x^9 - 8744103079*x^10 +...
such that
log(A(x)) = x - 3*x^2/2 + 19*x^3/3 - 207*x^4/4 + 3331*x^5/5 - 71223*x^6/6 + 1890379*x^7/7 - 59652687*x^8/8 + 2175761971*x^9/9 +...+ A180563(n)*x^n/n +...
where the e.g.f. G(x) of A180563 begins
G(x) = x - 3*x^2/2! + 19*x^3/3! - 207*x^4/4! + 3331*x^5/5! - 71223*x^6/6! + 1890379*x^7/7! +...+ A180563(n)*x^n/n! +...
and satisfies: Product_{n>=1} (1 - G(x)^n) = exp(-x).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A180653, A294331 (variant).

{A180563(n) = my( L = sum(m=1, n, sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(L), n)}
{a(n) = my(A); A = exp( sum(m=1, n+1, A180563(m)*x^m/m +x*O(x^n)) ); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

A180750 a(n) = DP(n) is the total number of k-double-palindromes of n, where 2 <= k <= n.

Original entry on oeis.org

0, 1, 3, 6, 13, 21, 43, 68, 116, 185, 311, 464, 757, 1157, 1741, 2720, 4081, 6214, 9199, 14078, 20353, 31405, 45035, 68930, 98224, 150761, 212706, 326362, 458725, 702209, 983011, 1504400, 2096441, 3207137, 4456139, 6808172, 9437149, 14408669, 19921297, 30393800

Offset: 1

John P. McSorley, Sep 19 2010

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 (see sequence A180653) 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.
The n-th term of this sequence is DP(n), the total number of k-double-palindromes of n, where 2 <= k <= n.
For example, DP(6)=21 because there are 21 k-double-palindromes of 6 for k=2,3,4,5, or 6. They are:
(with k=2) 15=1|5, 51=5|1, 24=2|4, 42=4|2, 33=3|3,
(with k=3) 114=11|4, 411=4|11, 222=2|22,
(with k=4) 1113=111|3, 3111=3|111, 1311=131|1, 1131=1|131, and 1122=11|22, 2211=22|11, 1212=121|2, 2121=2|121,
(with k=5) 11112=1111|2, 21111=2|1111, 12111=121|11, 11121=11|121,
(with k=6) 111111=1|11111.

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

a(n) is the sum of the n-th row of the triangle sequence A180653 (number of k-double-palindromes of n).

The n-th term of sequence A016116 is the total number of k-palindromes (single palindromes) of n.

a(n) = A180249(n) - A179781(n). - Petros Hadjicostas, Nov 03 2017

G.f.: Sum_{n>=1} phi^{(-1)}(n)*f(x^n) - Sum_{n>=1} mu(n)*g(x^n), where phi^{(-1)}(n) = A023900(n) is the Dirichlet inverse of Euler's totient function, mu(n) = A008683(n) is the Mobius function, f(x) = x*(x+1)*(2*x+1)/(1-2*x^2)^2, and g(x) = x*(1+2*x)/(1-2*x^2). - Petros Hadjicostas, Nov 06 2017

a(11)-a(18) from Donovan Johnson, Oct 22 2010

a(11)-a(18) corrected by and a(19)-a(40) from Petros Hadjicostas and Andrew Howroyd, Nov 03 2017

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A180918 'DPE(n,k)' triangle read by rows. DPE(n,k) is the number of k-double-palindromes of n up to cyclic equivalence.

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A181111 'ADP(n,k)' triangle read by rows. ADP(n,k) is the number of aperiodic k-double-palindromes of n.

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A294332 G.f.: exp( Sum_{n>=1} A180563(n) * x^n / n ).

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Author

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Programs

PARI

A180750 a(n) = DP(n) is the total number of k-double-palindromes of n, where 2 <= k <= n.

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Author

Keywords

Comments

References

Crossrefs

Formula

Extensions