OFFSET
0,2
COMMENTS
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.
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.
LINKS
D. A. Gregory, S. Kirkland, and N. J. Pullman, Power convergent Boolean matrices, Linear Algebra and its Applications, Volume 179, 15 January 1993, Pages 105-117.
E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
FORMULA
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).
MATHEMATICA
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Sep 10 2023
STATUS
approved