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.

Showing 1-3 of 3 results.

A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 1, 8, 10, 10, 1, 1, 1, 16, 22, 52, 26, 1, 1, 1, 32, 46, 232, 196, 76, 1, 1, 1, 64, 94, 976, 1016, 1216, 232, 1, 1, 1, 128, 190, 4000, 4576, 12496, 5944, 764, 1, 1, 1, 256, 382, 16192, 19376, 111376, 73648, 42400, 2620, 1
Offset: 0

Views

Author

Andrew Howroyd, May 06 2023

Keywords

Comments

Two involutions x,y on [n] commute if x*y = y*x.

Examples

			Array begins:
===========================================================
n/k| 0   1    2     3      4       5        6         7 ...
---+-------------------------------------------------------
0  | 1   1    1     1      1       1        1         1 ...
1  | 1   1    1     1      1       1        1         1 ...
2  | 1   2    4     8     16      32       64       128 ...
3  | 1   4   10    22     46      94      190       382 ...
4  | 1  10   52   232    976    4000    16192     65152 ...
5  | 1  26  196  1016   4576   19376    79696    323216 ...
6  | 1  76 1216 12496 111376  936976  7680016  62177296 ...
7  | 1 232 5944 73648 716416 6289312 52647904 430723168 ...
  ...
		

Crossrefs

Columns k=0..3 are A000012, A000085, A362819, A362825.
Rows n=2..3 are A000079, A033484.
Main diagonal is A362823.

Programs

  • PARI
    \\ B(n,k) is A022166.
    B(n,k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
    T(n,k)={if(n==0, 1, n!*polcoef(exp(sum(j=0, min(k,logint(n,2)), B(k,j)*x^(2^j)/2^j, O(x*x^n))), n))}

Formula

T(0,k) = T(1,k) = 1.

A362825 Number of ordered triples of involutions on [n] that pairwise commute.

Original entry on oeis.org

1, 1, 8, 22, 232, 1016, 12496, 73648, 1032032, 7586272, 118141696, 1033672256, 17668427008, 178649596672, 3313667912192, 37898019913216, 756948065453056, 9640771045925888, 205935949714235392, 2885307792776353792, 65568056040976818176
Offset: 0

Views

Author

Andrew Howroyd, May 06 2023

Keywords

Comments

Two involutions x,y on [n] commute if x*y = y*x.

Crossrefs

Column k=3 of A362824.
Cf. A362819.

Programs

  • PARI
    seq(n) = {Vec(serlaplace(exp(x + 7*x^2/2 + 7*x^4/4 + x^8/8 + O(x*x^n))))}

Formula

E.g.f.: exp(x + 7*x^2/2 + 7*x^4/4 + x^8/8).

A362820 Number of ordered pairs of derangements on [n] that commute.

Original entry on oeis.org

1, 0, 1, 4, 33, 136, 1825, 10956, 163009, 1575568, 23894721, 280090900, 5410068961, 73066199064, 1483125027553, 25872759745756, 561027082980225, 10796395534986016, 266457543316023169, 5743345672152317988, 152031229968147150241, 3717043193920429157800, 104377807879737865769121
Offset: 0

Views

Author

Andrew Howroyd, May 05 2023

Keywords

Comments

A derangement is a permutation without fixed points. Two permutations x,y commute if x*y = y*x.

Crossrefs

A053529 is the corresponding sequence for all permutations.

Programs

  • PARI
    seq(n)=Vec(serlaplace((1 - x)^2*exp(sum(k=1, n, (x^k/k)/(1-x^k) + O(x*x^n)) + x)))

Formula

E.g.f.: (1 - x)^2 * exp(x) * B(x) where B(x) is the e.g.f. of A053529.
Showing 1-3 of 3 results.