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

%I A362824 #8 May 08 2023 15:32:28
%S A362824 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,
%T A362824 46,232,196,76,1,1,1,64,94,976,1016,1216,232,1,1,1,128,190,4000,4576,
%U A362824 12496,5944,764,1,1,1,256,382,16192,19376,111376,73648,42400,2620,1
%N A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.
%C A362824 Two involutions x,y on [n] commute if x*y = y*x.
%H A362824 Andrew Howroyd, <a href="/A362824/b362824.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%F A362824 T(0,k) = T(1,k) = 1.
%e A362824 Array begins:
%e A362824 ===========================================================
%e A362824 n/k| 0   1    2     3      4       5        6         7 ...
%e A362824 ---+-------------------------------------------------------
%e A362824 0  | 1   1    1     1      1       1        1         1 ...
%e A362824 1  | 1   1    1     1      1       1        1         1 ...
%e A362824 2  | 1   2    4     8     16      32       64       128 ...
%e A362824 3  | 1   4   10    22     46      94      190       382 ...
%e A362824 4  | 1  10   52   232    976    4000    16192     65152 ...
%e A362824 5  | 1  26  196  1016   4576   19376    79696    323216 ...
%e A362824 6  | 1  76 1216 12496 111376  936976  7680016  62177296 ...
%e A362824 7  | 1 232 5944 73648 716416 6289312 52647904 430723168 ...
%e A362824   ...
%o A362824 (PARI) \\ B(n,k) is A022166.
%o A362824 B(n,k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
%o A362824 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))}
%Y A362824 Columns k=0..3 are A000012, A000085, A362819, A362825.
%Y A362824 Rows n=2..3 are A000079, A033484.
%Y A362824 Main diagonal is A362823.
%Y A362824 Cf. A022166, A362648.
%K A362824 nonn,tabl
%O A362824 0,9
%A A362824 _Andrew Howroyd_, May 06 2023