A329228
Triangle read by rows: T(n,k) is the number of digraphs on n unlabeled vertices such that every vertex has outdegree k, n >= 1, 0 <= k < n.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 13, 79, 13, 1, 1, 40, 1499, 1499, 40, 1, 1, 100, 35317, 257290, 35317, 100, 1, 1, 291, 967255, 56150820, 56150820, 967255, 291, 1, 1, 797, 29949217, 14971125930, 111359017198, 14971125930, 29949217, 797, 1
Offset: 1
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 6, 6, 1;
1, 13, 79, 13, 1;
1, 40, 1499, 1499, 40, 1;
1, 100, 35317, 257290, 35317, 100, 1;
1, 291, 967255, 56150820, 56150820, 967255, 291, 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}
E(v, x) = {my(r=1/(1-x)); for(i=1, #v, r=serconvol(r, prod(j=1, #v, my(g=gcd(v[i], v[j])); (1 + x^(v[j]/g))^g)/(1 + x))); r}
Row(n)={my(s=0); forpart(p=n, s+=permcount(p)*E(p, x+O(x^n))); Vec(s/n!)}
{ for(n=1, 8, print(Row(n))) }
A129524
Number of unlabeled digraphs on n vertices such that every vertex has outdegree 2.
Original entry on oeis.org
0, 0, 1, 6, 79, 1499, 35317, 967255, 29949217, 1033242585, 39323062014, 1637375262965, 74075329383599, 3619112881630497, 189953824713590782, 10661151595417930874, 637230479685691806302, 40415532825383300892418, 2711124591869919503655096
Offset: 1
A185194
Number of unlabeled digraphs on n vertices such that every vertex has outdegree 4.
Original entry on oeis.org
0, 0, 0, 0, 1, 40, 35317, 56150820, 111359017198, 278086517599356, 877760741062694898, 3482578978170418753715, 17204168691253789080138981, 104690934973509839187285618311, 776311587313178356520412354767734, 6942595716239018207126337605515965388
Offset: 1
A185303
Number of unlabeled digraphs on n vertices such that every vertex has outdegree 5.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 100, 967255, 14971125930, 278086517599356, 6521004095675547914, 197419530111112377546537, 7747427934648623352166753715, 392370903258277676503800999871543, 25436929780226775791085690703723141426, 2090584629532654146005764252197925046719651
Offset: 1
A005535
Number of semi-regular digraphs (with loops) on n unlabeled nodes with each node having out-degree 3.
Original entry on oeis.org
1, 19, 916, 91212, 12888450, 2411213698, 575737451509, 171049953499862, 61944438230597774, 26879022100485977540, 13773587720396658214925, 8231894671550187551622795, 5676740663627528580559535893, 4474748487205893704072253926113
Offset: 3
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
-
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, k_] := Product[SeriesCoefficient[Product[g = GCD[v[[i]], v[[j]]]; (1 + x^(v[[j]]/g) + O[x]^(k + 1))^g, {j, 1, Length[v]}], {x, 0, k}], {i, 1, Length[v]}];
a[n_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, 3], {p, IntegerPartitions[n]}]; s/n!];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 20 2022, after Andrew Howroyd in A259471 *)
Showing 1-5 of 5 results.