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A362903
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Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.
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2
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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
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OFFSET
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0,5
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).
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EXAMPLE
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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 ...
...
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PROG
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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, ])) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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