A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 11, 4, 1, 1, 16, 43, 24, 5, 1, 1, 32, 171, 176, 46, 6, 1, 1, 64, 683, 1376, 611, 80, 7, 1, 1, 128, 2731, 10944, 9281, 1864, 130, 8, 1, 1, 256, 10923, 87424, 146445, 54384, 5161, 200, 9, 1, 1, 512, 43691, 699136, 2334181, 1696352, 285939, 13184, 295, 10, 1

Offset

0,5

Comments

Two involutions x,y commute if x*y = y*x. Isomorphism is up to permutation of the elements of the (2n)-set. T(n,k) also gives the values for a (2n+1)-set.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Formula

G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).

Example

Array begins:
======================================================
n/k| 0 1   2     3       4          5            6 ...
---+--------------------------------------------------
0  | 1 1   1     1       1          1            1 ...
1  | 1 2   4     8      16         32           64 ...
2  | 1 3  11    43     171        683         2731 ...
3  | 1 4  24   176    1376      10944        87424 ...
4  | 1 5  46   611    9281     146445      2334181 ...
5  | 1 6  80  1864   54384    1696352     53885632 ...
6  | 1 7 130  5161  285939   17562679   1110290303 ...
7  | 1 8 200 13184 1372224  165343616  20774749952 ...
8  | 1 9 295 31532 6101080 1436647664 358238974304 ...
  ...

Prog

(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)}
C(k, n) = Vec(1/prod(j=0, min(k-1, logint(n, 2)), (1 - x^(2^j) + O(x*x^n))^B(k, j+1), 1 - x + O(x*x^n)))
M(n, m=n) = Mat(vector(m+1, k, C(k-1, n)~))
{ my(A=M(7)); for(i=1, #A, print(A[i, ])) }

Crossrefs

Columns k=0..3 are A000012, A000027(n-1), A001752, A362904.

Rows n=1..3 are A000079, A007583, A103334(n+1).

Cf. A022166, A362648, A362824, A362826.

Sequence in context: A103323 A329332 A092056 * A103574 A112682 A033185

Adjacent sequences: A362900 A362901 A362902 * A362904 A362905 A362906

Keyword

nonn,tabl

Author

Andrew Howroyd, May 11 2023

Status

approved