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A057273
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Triangle T(n,k) of the number of strongly connected digraphs on n labeled nodes and with k arcs, k=0..n*(n-1).
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13
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1, 0, 0, 1, 0, 0, 0, 2, 9, 6, 1, 0, 0, 0, 0, 6, 84, 316, 492, 417, 212, 66, 12, 1, 0, 0, 0, 0, 0, 24, 720, 6440, 26875, 65280, 105566, 122580, 106825, 71700, 37540, 15344, 4835, 1140, 190, 20, 1, 0, 0, 0, 0, 0, 0, 120, 6480, 107850, 868830, 4188696, 13715940
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OFFSET
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1,8
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REFERENCES
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Archer, K., Gessel, I. M., Graves, C., & Liang, X. (2020). Counting acyclic and strong digraphs by descents. Discrete Mathematics, 343(11), 112041. See Table 2.
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LINKS
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EXAMPLE
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Triangle begins:
[1] 1;
[2] 0,0,1;
[3] 0,0,0,2,9,6,1;
[4] 0,0,0,0,6,84,316,492,417,212,66,12,1;
...
Number of strongly connected digraphs on 3 labeled nodes is 18 = 2+9+6+1.
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PROG
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(PARI)
B(nn, e=2)={my(v=vector(nn)); for(n=1, nn, v[n] = e^(n*(n-1)) - sum(k=1, n-1, binomial(n, k)*e^((n-1)*(n-k))*v[k])); v}
Strong(n, e=2)={my(u=B(n, e), v=vector(n)); v[1]=1; for(n=2, #v, v[n] = u[n] + sum(j=1, n-1, binomial(n-1, j-1)*u[n-j]*v[j])); v}
row(n)={ Vecrev(Strong(n, 1+'y)[n]) }
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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