A360862
Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).
Original entry on oeis.org
1, 1, 2, 1, 4, 1, 7, 5, 1, 10, 20, 5, 1, 14, 48, 36, 1, 18, 99, 153, 30, 1, 23, 181, 481, 277, 17, 1, 28, 303, 1239, 1451, 323, 1, 34, 479, 2811, 5572, 2946, 193, 1, 40, 726, 5805, 17607, 17343, 3806, 71, 1, 47, 1055, 11148, 48401, 77708, 36872, 3188, 1, 54, 1492, 20219, 120018, 288476, 243007, 54386, 1496
Offset: 2
Triangle begins:
1;
1, 2;
1, 4;
1, 7, 5;
1, 10, 20, 5;
1, 14, 48, 36;
1, 18, 99, 153, 30;
1, 23, 181, 481, 277, 17;
1, 28, 303, 1239, 1451, 323;
1, 34, 479, 2811, 5572, 2946, 193;
1, 40, 726, 5805, 17607, 17343, 3806, 71;
1, 47, 1055, 11148, 48401, 77708, 36872, 3188;
1, 54, 1492, 20219, 120018, 288476, 243007, 54386, 1496;
...
A309936
Irregular triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs with n edges covering k vertices, n >= 1, 1 <= k <= 2*n.
Original entry on oeis.org
0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 1, 1, 0, 1, 3, 7, 6, 4, 1, 1, 0, 1, 4, 13, 17, 17, 8, 4, 1, 1, 0, 1, 6, 25, 44, 56, 41, 24, 9, 4, 1, 1, 0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 1, 0, 1, 9, 65, 216, 450, 562, 503, 315, 162, 64, 27, 9, 4, 1, 1
Offset: 1
Triangle begins:
0, 1;
0, 1, 1, 1;
0, 1, 2, 3, 1, 1;
0, 1, 3, 7, 6, 4, 1, 1;
0, 1, 4, 13, 17, 17, 8, 4, 1, 1;
0, 1, 6, 25, 44, 56, 41, 24, 9, 4, 1, 1;
0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
{ my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }
A327728
Number of unlabeled multigraphs with loops allowed and n edges covering three vertices.
Original entry on oeis.org
0, 2, 8, 19, 40, 77, 132, 217, 340, 510, 742, 1054, 1456, 1976, 2634, 3453, 4464, 5703, 7194, 8987, 11120, 13636, 16588, 20036, 24024, 28630, 33916, 39951, 46816, 54601, 63376, 73253, 84324, 96690, 110466, 125778, 142728, 161468, 182126, 204841, 229768, 257075, 286902, 319447, 354880, 393384
Offset: 1
a(2) = 2 since three vertices may be covered with two edges in 2 ways: the path graph P(3) or an edge plus a loop.
- Andrew Howroyd, Table of n, a(n) for n = 1..1000
- Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).
Showing 1-3 of 3 results.
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