Alberto Tacchella has authored 5 sequences.
A223256
Triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the numerator of the coefficient of x^k in the characteristic polynomial of the matrix realizing the transformation to Jacobi coordinates for a system of n particles on a line.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 11, 11, 1, 1, 25, 61, 25, 1, 1, 137, 379, 379, 137, 1, 1, 49, 667, 3023, 667, 49, 1, 1, 363, 529, 8731, 8731, 529, 363, 1, 1, 761, 46847, 62023, 270961, 62023, 46847, 761, 1, 1, 7129, 51011, 9161, 28525, 28525, 9161, 51011, 7129, 1
Offset: 0
Triangle begins:
1,
1, 1,
1, 3, 1,
1, 11, 11, 1,
1, 25, 61, 25, 1,
1, 137, 379, 379, 137, 1,
1, 49, 667, 3023, 667, 49, 1,
1, 363, 529, 8731, 8731, 529, 363, 1,
...
A223257
Triangle read by rows: T(0,0)=1; for n>=1 T(n,k) is the denominator of the coefficient of x^k in the characteristic polynomial of the matrix realizing the transformation to Jacobi coordinates for a system of n particles on a line.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 24, 12, 1, 1, 60, 120, 120, 60, 1, 1, 20, 180, 720, 180, 20, 1, 1, 140, 126, 1680, 1680, 126, 140, 1, 1, 280, 10080, 10080, 40320, 10080, 10080, 280, 1, 1, 2520, 10080, 1296, 3456, 3456, 1296, 10080, 2520, 1
Offset: 0
Triangle begins:
1,
1, 1,
1, 2, 1,
1, 6, 6, 1,
1, 12, 24, 12, 1,
1, 60, 120, 120, 60, 1,
1, 20, 180, 720, 180, 20, 1,
1, 140, 126, 1680, 1680, 126, 140, 1,
...
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.
Original entry on oeis.org
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
Offset: 0
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;
...
- 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 [math.CO], 2017; see Section 4.
- Brendan McKay and Adolfo Piperno, nauty and Traces. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
- B. D. McKay and A. Piperno, Practical Graph Isomorphism, II, J. Symbolic Computation 60 (2013), 94-112.
- Gordon Royle, Small Multigraphs.
- Gus Wiseman, Illustration of the 33 connected multigraphs counted in row 5.
Cf.
A000664,
A007718,
A036250,
A050535,
A191646,
A191970,
A275421,
A317672,
A322114,
A322133,
A322152.
-
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(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
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
A192517
Table read by antidiagonals: T(n,k) = number of multigraphs with n vertices and k edges, with no loops allowed (n >= 1, k >= 0).
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 3, 6, 4, 1, 0, 1, 1, 3, 7, 11, 5, 1, 0, 1, 1, 3, 8, 17, 18, 7, 1, 0, 1, 1, 3, 8, 21, 35, 32, 8, 1, 0, 1, 1, 3, 8, 22, 52, 76, 48, 10, 1, 0, 1, 1, 3, 8, 23, 60, 132, 149, 75, 12, 1, 0
Offset: 1
Table begins:
[1,0,0,0,0,0,0,0,0,...],
[1,1,1,1,1,1,1,1,1,...],
[1,1,2,3,4,5,7,8,10,...],
[1,1,3,6,11,18,32,48,75,...],
[1,1,3,7,17,35,76,149,291,...],
[1,1,3,8,21,52,132,313,741,...],
[1,1,3,8,22,60,173,471,1303,...],
[1,1,3,8,23,64,197,588,1806,...],
...
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 171.
-
\\ See A191646 for G function.
R(n)={Mat(vectorv(n, k, concat([1], G(k, n-1))))}
{ my(A=R(10)); for(n=1, #A, for(k=1, #A, print1(A[n,k], ", "));print) } \\ Andrew Howroyd, May 14 2018
A191970
Number of connected graphs with n edges with loops allowed.
Original entry on oeis.org
1, 2, 2, 6, 12, 33, 93, 287, 940, 3309, 12183, 47133, 190061, 796405, 3456405, 15501183, 71681170, 341209173, 1669411182, 8384579797, 43180474608, 227797465130, 1229915324579, 6790642656907, 38311482445514, 220712337683628, 1297542216770482, 7779452884747298
Offset: 0
a(1)=2: Either one node with the edge equal to a loop, or two nodes connected by the edge. a(2)=2: Either three nodes on a chain connected by the two edges, or two nodes connected by an edge, one node with a loop. Apparently multi-loops are not allowed (?). - _R. J. Mathar_, Jul 25 2017
Cf.
A000664,
A002905,
A007718,
A050535,
A053419,
A054923,
A191646,
A191970,
A275421,
A322133,
A322151,
A322152.
-
\\ See A322114 for InvEulerMT, G.
seq(n)={vecsum([Vec(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])} \\ Andrew Howroyd, Oct 22 2019
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