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A327377
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Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly k endpoints (vertices of degree 1).
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6
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1, 0, 0, 0, 0, 1, 1, 0, 3, 0, 10, 12, 12, 4, 3, 253, 260, 160, 60, 35, 0, 12068, 9150, 4230, 1440, 480, 66, 15, 1052793, 570906, 195048, 53200, 12600, 2310, 427, 0, 169505868, 63523656, 15600032, 3197040, 585620, 95088, 14056, 1016, 105
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OFFSET
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0,9
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COMMENTS
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A graph is covering if there are no isolated vertices.
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LINKS
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FORMULA
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Column-wise inverse binomial transform of A327369.
E.g.f.: exp(-x)*exp(x + U(x,y) + B(x*(1-y) + R(x,y))), where R(x,y) is the e.g.f. of A055302, U(x,y) is the e.g.f. of A055314 and B(x) + x is the e.g.f. of A059167. - Andrew Howroyd, Oct 05 2019
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EXAMPLE
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Triangle begins:
1
0 0
0 0 1
1 0 3 0
10 12 12 4 3
253 260 160 60 35 0
12068 9150 4230 1440 480 66 15
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PROG
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(PARI)
my(U=sum(n=2, n, x^n*sum(k=1, n, stirling(n-2, n-k, 2)*y^k/k!)) + O(x*x^n));
my(R=sum(n=1, n, x^n*sum(k=1, n, stirling(n-1, n-k, 2)*y^k/k!)) + O(x*x^n));
my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));
my(A=exp(-x + O(x*x^n))*exp(x + U + subst(B-x, x, x*(1-y) + R)));
Vecrev(n!*polcoef(A, n), n + 1);
}
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CROSSREFS
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Row sums without the first column are A327227.
The non-covering version is A327369.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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