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Triangle read by rows: T(n,k) is the number of strongly connected oriented graphs on n labeled nodes with k arcs, n >= 1, k=0..n*(n-1)/2.
5

%I #11 Jan 16 2022 17:46:09

%S 1,0,0,0,0,0,2,0,0,0,0,6,36,24,0,0,0,0,0,24,480,1940,2970,2040,544,0,

%T 0,0,0,0,0,120,5040,51330,221910,527940,772080,722250,426420,146160,

%U 22320,0,0,0,0,0,0,0,720,52920,1026060,8810970,43268442,138984510

%N Triangle read by rows: T(n,k) is the number of strongly connected oriented graphs on n labeled nodes with k arcs, n >= 1, k=0..n*(n-1)/2.

%H Andrew Howroyd, <a href="/A350731/b350731.txt">Table of n, a(n) for n = 1..1350</a> (rows 1..20)

%e Triangle begins:

%e [1] 1;

%e [2] 0, 0;

%e [3] 0, 0, 0, 2;

%e [4] 0, 0, 0, 0, 6, 36, 24;

%e [5] 0, 0, 0, 0, 0, 24, 480, 1940, 2970, 2040, 544;

%e ...

%o (PARI)

%o OrientedGgf(n, y=1) = {sum(k=0, n, ((1+2*y)/(1+y))^(k*(k-1)/2)*x^k/k!, O(x*x^n) )}

%o StrongO(n, y=1) = {my(g=serconvol(1/OrientedGgf(n,y), sum(k=0, n, x^k*(1+y)^(k*(k-1)/2), O(x*x^n)))); Vec(serlaplace(-log(g)))}

%o row(n)={Vecrev(StrongO(n,'y)[n], n*(n-1)/2+1)}

%o { for(n=1, 6, print(row(n))) }

%Y Row sums are A350730.

%Y The unlabeled version is A350750.

%Y Cf. A057273 (digraphs), A350732 (weakly connected).

%K nonn,tabf

%O 1,7

%A _Andrew Howroyd_, Jan 11 2022