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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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%I #10 Oct 06 2019 17:43:20

%S 1,0,0,0,0,1,1,0,3,0,10,12,12,4,3,253,260,160,60,35,0,12068,9150,4230,

%T 1440,480,66,15,1052793,570906,195048,53200,12600,2310,427,0,

%U 169505868,63523656,15600032,3197040,585620,95088,14056,1016,105

%N 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).

%C A graph is covering if there are no isolated vertices.

%H Andrew Howroyd, <a href="/A327377/b327377.txt">Table of n, a(n) for n = 0..1325</a>

%F Column-wise inverse binomial transform of A327369.

%F 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

%e Triangle begins:

%e 1

%e 0 0

%e 0 0 1

%e 1 0 3 0

%e 10 12 12 4 3

%e 253 260 160 60 35 0

%e 12068 9150 4230 1440 480 66 15

%o (PARI)

%o Row(n)={ \\ R, U, B are e.g.f. of A055302, A055314, A059167.

%o 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));

%o 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));

%o my(B=x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!)));

%o my(A=exp(-x + O(x*x^n))*exp(x + U + subst(B-x, x, x*(1-y) + R)));

%o Vecrev(n!*polcoef(A, n), n + 1);

%o }

%o { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Oct 05 2019

%Y Row sums are A006129.

%Y Column k = 0 is A100743.

%Y Column k = n is A123023(n + 2).

%Y Row sums without the first column are A327227.

%Y The non-covering version is A327369.

%Y The unlabeled version is A327372.

%Y Cf. A004110, A006125, A055302, A055314, A059167, A245797, A327362, A327364, A327370.

%K nonn,tabl

%O 0,9

%A _Gus Wiseman_, Sep 05 2019

%E Terms a(28) and beyond from _Andrew Howroyd_, Oct 05 2019