cp's OEIS Frontend

A365590 Number of n X n Boolean relation matrices such that each of the diagonal blocks of its Frobenius normal form is either a 1 block or a 0 block.

%I A365590 #16 Sep 11 2023 11:24:57
%S A365590 1,2,13,243,11998,1477763,436610299,300960642300,474171878424571,
%T A365590 1680899431189662775,13241419272545722904788,
%U A365590 229482664065433754849099977,8677282817864146616211588609715,710901968198799834001047038898570250
%N A365590 Number of n X n Boolean relation matrices such that each of the diagonal blocks of its Frobenius normal form is either a 1 block or a 0 block.
%C A365590  A 1(0) block is such that every entry in the block is 1(0).  See Gregory, Kirkland, Pullman for a description of Frobenius normal form.
%C A365590  a(n) is also the number of labeled digraphs (with loops allowed A002416) on [n] such that every strongly connected component is either complete or a single vertex without a loop.
%H A365590 D. A. Gregory, S. Kirkland, and N. J. Pullman, <a href="https://doi.org/10.1016/0024-3795(93)90323-G">Power convergent Boolean matrices</a>, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
%H A365590 E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%F A365590 Sum_{n>=0} a(n)*x^n/(n!*2^binomial(n,2)) = 1/(E(x) @ exp(-(exp(x)-1+x))) where E(x)=Sum_{n>=0} x^n/(n!*2^binomial(n,2)) and @ is the exponential Hadamard product (see Panafieu and Dovgal).
%t A365590 nn = 13; B[n_] := n! 2^Binomial[n, 2]; ggf[egf_] := Normal[Series[egf, {x, 0, nn}]] /. Table[x^i -> x^i/2^Binomial[i, 2], {i, 0, nn}]; Table[B[n], {n, 0, nn}] CoefficientList[Series[1/ggf[Exp[-(Exp[x] - 1 + x)]], {x, 0, nn}], x]
%Y A365590 Cf. A355612, A365593, A365534.
%K A365590 nonn
%O A365590 0,2
%A A365590 _Geoffrey Critzer_, Sep 10 2023