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%I #12 Aug 27 2019 10:44:34
%S 1,0,1,0,2,1,0,6,3,1,0,21,29,4,1,0,112,2101,150,5,1,0,853,7011181,
%T 7013164,1037,6,1,0,11117,1788775603301,29281354507753847,
%U 1788782615612,12338,7,1,0,261080,53304526022885278403,234431745534048893449761040648508,234431745534048922729326772799024,53304527811667884902,274659,8,1
%N Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.
%F Column k is the inverse Euler transform of column k of A301922. - _Andrew Howroyd_, Aug 26 2019
%e Triangle begins:
%e 1
%e 0 1
%e 0 2 1
%e 0 6 3 1
%e 0 21 29 4 1
%e 0 112 2101 150 5 1
%e 0 853 7011181 7013164 1037 6 1
%e ...
%e The T(4,2) = 6 hypergraphs:
%e {{1,3},{2,4},{3,4}}
%e {{1,4},{2,4},{3,4}}
%e {{1,2},{1,3},{2,4},{3,4}}
%e {{1,4},{2,3},{2,4},{3,4}}
%e {{1,3},{1,4},{2,3},{2,4},{3,4}}
%e {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}
%o (PARI)
%o InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}
%o 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}
%o rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L<k, listput(L, #L))); Vec(L)}
%o can(v, f)={my(d=1, u=v); while(d>0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
%o Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
%o U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}
%o A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}
%o { my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ _Andrew Howroyd_, Aug 26 2019
%Y Row sums are A301920.
%Y Columns k=2..3 are A001349(n > 1), A003190(n > 1).
%Y Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.
%K nonn,tabl
%O 1,5
%A _Gus Wiseman_, Jun 19 2018
%E Terms a(16) and beyond from _Andrew Howroyd_, Aug 26 2019
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