A361269 - OEIS

%I #28 Mar 16 2023 04:50:52

%S 1,0,2,0,4,12,0,144,168,200,0,25696,18768,12384,8688,0,18082560,

%T 8697280,3923040,1914560,936992,0,47025585664,14670384000,4512045120,

%U 1622358720,647087040,242016192,0,450955726792704,87781550054912,17679638000640,4496696041600,1408276410240,482302375296,145763745920

%N Triangular array read by rows. T(n,k) is the number of binary relations on [n] containing exactly k strongly connected components, n >= 0, 0 <= k <= n.

%H 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.

%H R. W. Robinson, <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.pdf">Counting digraphs with restrictions on the strong components</a>, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Strongly_connected_component">Strongly connected component</a>

%F E.g.f. for column 1: A(2*x) where A(x) is the e.g.f. for A003030.

%F E.g.f. for main diagonal: B(2*x) where B(x) is the e.g.f. for A003024.

%e   1;

%e   0,     2;

%e   0,     4,    12;

%e   0,   144,   168,   200;

%e   0, 25696, 18768, 12384, 8688;

%e   ...

%t nn =15; strong = Select[Import["https://oeis.org/A003030/b003030.txt", "Table"],

%t    Length@# == 2 &][[All, 2]]; s[x_] := Total[strong Table[x^i/i!, {i, 1, 58}]]; begf = Total[CoefficientList[ Series[1/(Total[CoefficientList[Series[ Exp[-u *s[x]], {x, 0, nn}], x]* Table[z^n/(2^Binomial[n, 2]), {n, 0, nn}]]), {z, 0, nn}],z]*Table[z^n 2^Binomial[n, 2], {n, 0, nn}]] /. z -> 2 z;

%t Range[0, nn]! CoefficientList[begf, {z, u}] // Grid (* _Geoffrey Critzer_, Mar 14 2023 after Andrew Howroyd *)

%o (PARI)

%o Z(p, f)={my(n=serprec(p, x)); serconvol(p, sum(k=0, n-1, x^k*f(k), O(x^n)))}

%o G(e, p)={Z(p, k->1/e^(k*(k-1)/2))}

%o U(e, p)={Z(p, k->e^(k*(k-1)/2))}

%o RelEgf(n, e)={sum(k=0, n, e^(k^2)*x^k/k!, O(x*x^n) )}

%o T(n)={my(e=2); [Vecrev(p) | p<-Vec(serlaplace(U(e, 1/G(e, exp(y*log(U(e, 1/G(e, RelEgf(n, e)))))))))]}

%o { my(A=T(6)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Mar 06 2023

%Y Cf. A003030, A003024, A002416 (row sums).

%K nonn,tabl

%O 0,3

%A _Geoffrey Critzer_, Mar 06 2023

%E Terms a(15) and beyond from _Andrew Howroyd_, Mar 06 2023