A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 21, 29, 4, 1, 0, 112, 2101, 150, 5, 1, 0, 853, 7011181, 7013164, 1037, 6, 1, 0, 11117, 1788775603301, 29281354507753847, 1788782615612, 12338, 7, 1, 0, 261080, 53304526022885278403, 234431745534048893449761040648508, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1

Offset: 1

Author

Gus Wiseman, Jun 19 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   1
   0    1
   0    2       1
   0    6       3       1
   0   21      29       4    1
   0  112    2101     150    5 1
   0  853 7011181 7013164 1037 6 1
   ...
The T(4,2) = 6 hypergraphs:
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
		

Crossrefs

Row sums are A301920.

Columns k=2..3 are A001349(n > 1), A003190(n > 1).

Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.

Programs

PARI

InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}
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}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}
A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}
{ my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 26 2019

Formula

Column k is the inverse Euler transform of column k of A301922. - Andrew Howroyd, Aug 26 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019