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).

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, 126, 40, 20, 6, 4, 0, 8047, 2934, 968, 263, 92, 25, 13, 3, 1, 205914, 53768, 11752, 2434, 596, 140, 47, 12, 5, 0, 10014882, 1707627, 240615, 34756, 5864, 1084, 256, 58, 21, 4, 1

Offset

0,7

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Gus Wiseman, The graphs counted in row 5 (isolated vertices not shown).

Formula

Column-wise partial sums of A327372.

Example

Triangle begins:
     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, 126,  40, 20,  6,  4, 0;
  8047, 2934, 968, 263, 92, 25, 13, 3, 1;
  ...

Prog

(PARI)
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}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
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)}
T(n)={my(v=Vec(G(n))); vector(#v, n, Vecrev(v[n], n))}
my(A=T(10)); for(n=1, #A, print(A[n])) \\ Andrew Howroyd, Jan 22 2021

Crossrefs

Row sums are A000088.

Row sums without the first column are A141580.

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

Column k = n is A059841.

Column k = n - 1 is A028242.

The labeled version is A327369.

The covering case is A327372.

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

Sequence in context: A011420 A035686 A308214 * A037228 A002117 A042970

Adjacent sequences: A327368 A327369 A327370 * A327372 A327373 A327374

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 04 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Sep 05 2019

Status

approved