A327371 - OEIS

%I #15 Jan 22 2021 16:48:09

%S 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 126,40,20,6,4,0,8047,2934,968,263,92,25,13,3,1,205914,53768,11752,

%U 2434,596,140,47,12,5,0,10014882,1707627,240615,34756,5864,1084,256,58,21,4,1

%N 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 Andrew Howroyd, <a href="/A327371/b327371.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H Gus Wiseman, <a href="/A327371/a327371.png">The graphs counted in row 5 (isolated vertices not shown).</a>

%F Column-wise partial sums of A327372.

%e Triangle begins:

%e      1;

%e      1,    0;

%e      1,    0,   1;

%e      2,    0,   2,   0;

%e      5,    1,   3,   1,  1;

%e     16,    6,   7,   2,  3,  0;

%e     78,   35,  25,   8,  7,  2,  1;

%e    588,  260, 126,  40, 20,  6,  4, 0;

%e   8047, 2934, 968, 263, 92, 25, 13, 3, 1;

%e   ...

%o (PARI)

%o 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 edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}

%o 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 T(n)={my(v=Vec(G(n))); vector(#v, n, Vecrev(v[n], n))}

%o my(A=T(10)); for(n=1, #A, print(A[n])) \\ _Andrew Howroyd_, Jan 22 2021

%Y Row sums are A000088.

%Y Row sums without the first column are A141580.

%Y Columns k = 0..2 are A004110, A325115, A325125.

%Y Column k = n is A059841.

%Y Column k = n - 1 is A028242.

%Y The labeled version is A327369.

%Y The covering case is A327372.

%Y Cf. A055540, A059167, A245797, A294217, A327227, A327370.

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Sep 04 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Sep 05 2019