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A327371 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).

%I A327371 #15 Jan 22 2021 16:48:09
%S A327371 1,1,0,1,0,1,2,0,2,0,5,1,3,1,1,16,6,7,2,3,0,78,35,25,8,7,2,1,588,260,
%T A327371 126,40,20,6,4,0,8047,2934,968,263,92,25,13,3,1,205914,53768,11752,
%U A327371 2434,596,140,47,12,5,0,10014882,1707627,240615,34756,5864,1084,256,58,21,4,1
%N A327371 Triangle read by rows where T(n,k) is the number of unlabeled simple graphs with n vertices and exactly k endpoints (vertices of degree 1).
%H A327371 Andrew Howroyd, <a href="/A327371/b327371.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A327371 Gus Wiseman, <a href="/A327371/a327371.png">The graphs counted in row 5 (isolated vertices not shown).</a>
%F A327371 Column-wise partial sums of A327372.
%e A327371 Triangle begins:
%e A327371      1;
%e A327371      1,    0;
%e A327371      1,    0,   1;
%e A327371      2,    0,   2,   0;
%e A327371      5,    1,   3,   1,  1;
%e A327371     16,    6,   7,   2,  3,  0;
%e A327371     78,   35,  25,   8,  7,  2,  1;
%e A327371    588,  260, 126,  40, 20,  6,  4, 0;
%e A327371   8047, 2934, 968, 263, 92, 25, 13, 3, 1;
%e A327371   ...
%o A327371 (PARI)
%o A327371 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A327371 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
%o A327371 G(n)={sum(k=0, n, my(s=0); forpart(p=k, s+=permcount(p) * 2^edges(p) * prod(i=1, #p, (1 - x^p[i])/(1 - (x*y)^p[i]) + O(x*x^(n-k)))); x^k*s/k!)*(1-x^2*y)/(1-x^2*y^2)}
%o A327371 T(n)={my(v=Vec(G(n))); vector(#v, n, Vecrev(v[n], n))}
%o A327371 my(A=T(10)); for(n=1, #A, print(A[n])) \\ _Andrew Howroyd_, Jan 22 2021
%Y A327371 Row sums are A000088.
%Y A327371 Row sums without the first column are A141580.
%Y A327371 Columns k = 0..2 are A004110, A325115, A325125.
%Y A327371 Column k = n is A059841.
%Y A327371 Column k = n - 1 is A028242.
%Y A327371 The labeled version is A327369.
%Y A327371 The covering case is A327372.
%Y A327371 Cf. A055540, A059167, A245797, A294217, A327227, A327370.
%K A327371 nonn,tabl
%O A327371 0,7
%A A327371 _Gus Wiseman_, Sep 04 2019
%E A327371 Terms a(21) and beyond from _Andrew Howroyd_, Sep 05 2019