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Triangle read by rows: T(n,k) is the number of simple digraphs on labeled n nodes with k strongly connected components.
3

%I #10 May 04 2023 14:57:12

%S 1,0,1,0,1,3,0,18,21,25,0,1606,1173,774,543,0,565080,271790,122595,

%T 59830,29281,0,734774776,229224750,70500705,25349355,10110735,3781503,

%U 0,3523091615568,685793359804,138122171880,35130437825,11002159455,3767987307,1138779265

%N Triangle read by rows: T(n,k) is the number of simple digraphs on labeled n nodes with k strongly connected components.

%H Andrew Howroyd, <a href="/A361455/b361455.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50).

%F T(n,k) = A361269(n,k)/2^n.

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 3;

%e 0, 18, 21, 25;

%e 0, 1606, 1173, 774, 543;

%e 0, 565080, 271790, 122595, 59830, 29281;

%e 0, 734774776, 229224750, 70500705, 25349355, 10110735, 3781503;

%e ...

%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 DigraphEgf(n, e)={sum(k=0, n, e^(k*(k-1))*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, DigraphEgf(n, e)))))))))]}

%o { my(A=T(6)); for(i=1, #A, print(A[i])) }

%Y Column k=1 is A003030.

%Y Main diagonal is A003024.

%Y Row sums are A053763.

%Y The unlabeled version is A361582.

%Y Cf. A189898 (weak components), A361269 (loops allowed), A361591.

%K nonn,tabl

%O 0,6

%A _Andrew Howroyd_, Mar 16 2023