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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). 9
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified April 18 02:38 EDT 2021. Contains 343072 sequences. (Running on oeis4.)