OFFSET
0,2
COMMENTS
The sequence gives a good lower bound for the number of convergent binary relations (A365534) which is only known for n <= 6.
LINKS
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(-(sm(x)-1+x))) where E(x) = Sum_{n>=0} x^n/(n!*2^binomial(n,2)), sm(x) = Sum_{n>=0} (2^n-1)*A003030(n)*x^n/n! and @ is the exponential Hadamard product (see Panafieu and Dovgal).
EXAMPLE
a(2) = 15 because there are 16 labeled digraphs with self loops on [2] and all of them are good except: [1->2,2->1].
MATHEMATICA
nn = 12; B[n_] := 2^Binomial[n, 2] n!; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"], Length@# == 2 &][[All, 2]]; sm[x_] := Total[Table[2^n - 1, {n, 1, Length[strong]}] strong Table[ x^i/i!, {i, 1, 58}]]; 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[-(sm[x] + x)]], {x, 0, nn}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Oct 19 2023
STATUS
approved