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A304074 Number of simple connected graphs with n nodes rooted at a pair of distinguished vertices. 7
0, 1, 4, 23, 162, 1549, 21090, 446061, 15673518, 961338288, 105752617892, 21155707801451, 7757777336382702, 5245054939576054088, 6571185585793205495484, 15325133281701584879975433, 66813349775478836190531605234, 546646811841381587823502759339055 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..50

FORMULA

a(n) = A304072(n) + A304073(n).

G.f.: 2*B(x)/G(x) - (x*C(x)/G(x))^2, where B(x) is the g.f. of A304069, C(x) is the g.f. of A000666 and G(x) is the g.f. of A000088. - Andrew Howroyd, Sep 07 2019

EXAMPLE

a(3)=4: one choice to mark two roots in the triangular graph; one choice to mark the two leaves in the linear graph; two choices to mark the center node and a leave (1st root in the center or 2nd root in the center) in the linear graph.

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)}

cross(u, v) = {sum(i=1, #u, sum(j=1, #v, gcd(u[i], v[j])))}

S(n, r)={my(t=#r+1); vector(n+1, n, if(n<t, 0, my(s=0); forpart(p=n-t, s+=permcount(p)*(2^(edges(p))*(2^cross(r, p)))); s/(n-t)!))}

seq(n)={my(g=Ser(S(n, []))); Vec(2*Ser(S(n, [1, 1]))/g - (Ser(S(n, [1]))/g)^2, -n)} \\ Andrew Howroyd, Sep 07 2019

CROSSREFS

Cf. A001349 (not rooted), A303831 (vertices not distinguished), A304070 (not necessarily connected).

Cf. A000088, A000666.

Sequence in context: A263186 A245110 A342988 * A307402 A111547 A171992

Adjacent sequences:  A304071 A304072 A304073 * A304075 A304076 A304077

KEYWORD

nonn

AUTHOR

Brendan McKay, May 05 2018

EXTENSIONS

Terms a(13) and beyond from Andrew Howroyd, Sep 07 2019

STATUS

approved

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Last modified January 28 12:53 EST 2022. Contains 350656 sequences. (Running on oeis4.)