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A191646 Triangle read by rows: T(n,k) = number of connected multigraphs with n>=0 edges and 1<=k<=n+1 vertices, with no loops allowed. 5
1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 11, 11, 6, 0, 1, 6, 22, 34, 29, 11, 0, 1, 7, 37, 85, 110, 70, 23, 0, 1, 9, 61, 193, 348, 339, 185, 47, 0, 1, 11, 95, 396, 969, 1318, 1067, 479, 106, 0, 1, 13, 141, 771, 2445, 4457, 4940, 3294, 1279, 235 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (terms 0..119 from R. J. Mathar)

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017), Section 4.

Gordon Royle, Small Multigraphs

Numbers in this table were computed using nauty 2.4

EXAMPLE

Triangle begins:

1;

0,1;

0,1,1;

0,1,2,2;

0,1,3,5,3;

0,1,4,11,11,6;

0,1,6,22,34,29,11;

...

PROG

(PARI)

EulerT(v)={my(p=exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1); Vec(p/x, -#v)}

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i) )}

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, x)={sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i], v[j])); g*x^(v[i]*v[j]/g))) + sum(i=1, #v, my(t=v[i]); ((t-1)\2)*x^t + if(t%2, 0, x^(t/2)))}

G(n, m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p, x) + O(x*x^m), -m))); s/n!}

R(n)={Mat(apply(p->Col(p+O(y^n), -n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k, n-1), y)))))}

{ my(A=R(10)); for(n=1, #A, for(k=1, n, print1(A[n, k], ", ")); print) } \\ Andrew Howroyd, May 14 2018

CROSSREFS

Cf. A054923, A192517, A253186 (column k=3), A290778 (column k=4).

Row sums give A076864.

Diagonal is A000055.

Sequence in context: A296068 A144064 A172236 * A297321 A277938 A130020

Adjacent sequences:  A191643 A191644 A191645 * A191647 A191648 A191649

KEYWORD

nonn,tabl,changed

AUTHOR

Alberto Tacchella, Jul 04 2011

STATUS

approved

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Last modified May 20 10:07 EDT 2018. Contains 304317 sequences. (Running on oeis4.)