A191646 revision #61
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A191646
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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.
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27
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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)
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OFFSET
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0,9
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LINKS
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FORMULA
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EXAMPLE
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Triangle T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
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;
...
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PROG
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(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
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CROSSREFS
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Cf. A000664, A007718, A036250, A050535, A191646, A191970, A275421, A317672, A322114, A322133, A322152.
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KEYWORD
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AUTHOR
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STATUS
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editing
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