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Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).
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%I #16 Oct 06 2019 17:46:19

%S 1,1,0,1,0,1,2,0,6,0,15,12,30,4,3,314,320,260,80,50,0,13757,10890,

%T 5445,1860,735,66,15,1142968,640836,228564,64680,16800,2772,532,0,

%U 178281041,68362504,17288852,3666600,702030,115416,17892,1016,105

%N Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

%H Andrew Howroyd, <a href="/A327369/b327369.txt">Table of n, a(n) for n = 0..1325</a>

%F Column-wise binomial transform of A327377.

%F E.g.f.: 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 1 0

%e 1 0 1

%e 2 0 6 0

%e 15 12 30 4 3

%e 314 320 260 80 50 0

%e 13757 10890 5445 1860 735 66 15

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Count[Length/@Split[Sort[Join@@#]],1]==k&]],{n,0,5},{k,0,n}]

%o (PARI)

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

%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(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(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 + 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 A006125.

%Y Row sums without the first column are A245797.

%Y Column k = 0 is A059167.

%Y Column k = 1 is A277072.

%Y Column k = 2 is A277073.

%Y Column k = 3 is A277074.

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

%Y Column k = n - 1 is A327370.

%Y The unlabeled version is A327371.

%Y The covering version is A327377.

%Y Cf. A004110, A055302, A055314, A059166, A059167, A100743, A327227, A327228, A327362, A327364.

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Sep 04 2019

%E Terms a(28) and beyond from _Andrew Howroyd_, Sep 09 2019