cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A284826 Irregular triangle T(n,k) for 1 <= k <= (n+1)/2: T(n,k) = number of primitive (aperiodic) palindromic structures of length n using exactly k different symbols.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 3, 1, 0, 2, 1, 0, 7, 6, 1, 0, 6, 6, 1, 0, 14, 25, 10, 1, 0, 12, 24, 10, 1, 0, 31, 90, 65, 15, 1, 0, 27, 89, 65, 15, 1, 0, 63, 301, 350, 140, 21, 1, 0, 56, 295, 349, 140, 21, 1, 0, 123, 965, 1701, 1050, 266, 28, 1
Offset: 1

Views

Author

Andrew Howroyd, Apr 03 2017

Keywords

Comments

Permuting the symbols will not change the structure.

Examples

			Triangle starts:
1
0
0   1
0   1
0   3    1
0   2    1
0   7    6     1
0   6    6     1
0  14   25    10     1
0  12   24    10     1
0  31   90    65    15     1
0  27   89    65    15     1
0  63  301   350   140    21    1
0  56  295   349   140    21    1
0 123  965  1701  1050   266   28   1
0 120  960  1700  1050   266   28   1
0 255 3025  7770  6951  2646  462  36  1
0 238 2999  7760  6950  2646  462  36  1
0 511 9330 34105 42525 22827 5880 750 45 1
0 495 9305 34095 42524 22827 5880 750 45 1
--------------------------------------------
For n=5, structures with 2 symbols are aabaa, ababa and abbba, so T(5,2) = 3.
For n=6, structures with 2 symbols are aabbaa and abbbba, so T(6,2) = 2.
(In this case, the structure abaaba is excluded because it is not primitive.)

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

Crossrefs

Columns 2-6 are A056481, A056482, A056483, A056484, A056485.
Partial row sums include A056476, A056477, A056478, A056479, A056480.
Row sums are A284841.
Cf. A284823.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*StirlingS2[Ceiling[#/2], k]&];
Table[T[n, k], {n, 1, 15}, {k, 1, Floor[(n+1)/2]}] // Flatten (* Jean-François Alcover, Jun 12 2017, from 2nd formula *)
  • PARI
    b(n,k) = sumdiv(n,d, moebius(n/d) * k^(ceil(d/2)));
a(n,k) = sum(j=0,k, b(n,k-j)*binomial(k,j)*(-1)^j)/k!;
for(n=1, 20, for(k=1, ceil(n/2), print1( a(n,k),", ");); print(););

Formula

T(n, k) = (Sum_{j=0..k} (-1)^j * binomial(k, j) * A284823(n, k-j)) / k!.
T(n, k) = Sum_{d | n} mu(n/d) * stirling2(ceiling(d/2), k).

A284823 Array read by antidiagonals: T(n,k) = number of primitive (aperiodic) palindromes of length n using a maximum of k different symbols (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 4, 0, 2, 0, 5, 0, 6, 2, 0, 6, 0, 12, 6, 6, 0, 7, 0, 20, 12, 24, 4, 0, 8, 0, 30, 20, 60, 18, 14, 0, 9, 0, 42, 30, 120, 48, 78, 12, 0, 10, 0, 56, 42, 210, 100, 252, 72, 28, 0, 11, 0, 72, 56, 336, 180, 620, 240, 234, 24, 0, 12, 0, 90, 72, 504, 294, 1290, 600, 1008, 216, 62
Offset: 1

Views

Author

Andrew Howroyd, Apr 03 2017

Keywords

Examples

			Table starts:
1  2   3    4    5    6     7     8     9    10 ...
0  0   0    0    0    0     0     0     0     0 ...
0  2   6   12   20   30    42    56    72    90 ...
0  2   6   12   20   30    42    56    72    90 ...
0  6  24   60  120  210   336   504   720   990 ...
0  4  18   48  100  180   294   448   648   900 ...
0 14  78  252  620 1290  2394  4088  6552  9990 ...
0 12  72  240  600 1260  2352  4032  6480  9900 ...
0 28 234 1008 3100 7740 16758 32704 58968 99900 ...
0 24 216  960 3000 7560 16464 32256 58320 99000 ...
...
Row 4 includes palindromes of the form abba but excludes those of the form aaaa, so T(4,k) is k*(k-1).
Row 6 includes palindromes of the forms aabbaa, abbbba, abccba but excludes those of the forms aaaaaa, abaaba, so T(6,k) is 2*k*(k-1) + k*(k-1)*(k-2).

References

  • M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Links

Crossrefs

Columns 2-6 are A056458, A056459, A056460, A056461, A056462.
Rows 5-10 are A007531(k+1), A045991, A058895, A047928(k-1), A135497, A133754.
Cf. A284826, A284841.

Programs

  • Mathematica
    T[n_, k_] := DivisorSum[n, MoebiusMu[n/#]*k^Ceiling[#/2]&]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jun 05 2017 *)
  • PARI
    a(n,k) = sumdiv(n, d, moebius(n/d) * k^(ceil(d/2)));
for(n=1, 10, for(k=1, 10, print1( a(n,k),", ");); print();)

Formula

T(n,k) = Sum_{d | n} mu(n/d) * k^(ceiling(d/2)).

A285013 Number of periodic palindromic structures of length n using an infinite alphabet.

Original entry on oeis.org

1, 1, 2, 2, 5, 5, 13, 15, 41, 52, 144, 203, 578, 877, 2605, 4140, 12869, 21147, 69178, 115975, 398766, 678570, 2450406, 4213597, 15939952, 27644437, 109304914, 190899322, 787016238, 1382958545, 5931824093, 10480142147, 46673259309, 82864869804, 382473282504
Offset: 0

Views

Author

Andrew Howroyd, Apr 07 2017

Keywords

Comments

See A285012 for additional information. Permuting the symbols will not change the structure.

Links

Crossrefs

Row sums of A285012.
Cf. A165137, A284841, A285042.

Programs

  • PARI
    \\ Requires T from A285012.
seq(n)={my(A=T(n)); concat([1], vector(n, i, vecsum(A[i,])))} \\ Andrew Howroyd, Oct 02 2019

Extensions

a(0)=1 prepended and terms a(28) and beyond from Andrew Howroyd, Oct 02 2019
Showing 1-3 of 3 results.

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