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 A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS FORMULA Column k is the inverse Euler transform of column k of A301922. - Andrew Howroyd, Aug 26 2019 EXAMPLE 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}} PROG (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 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. Sequence in context: A066387 A180663 A331327 * A262071 A011312 A275328 Adjacent sequences:  A301921 A301922 A301923 * A301925 A301926 A301927 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Jun 19 2018 EXTENSIONS Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019 STATUS approved

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Last modified February 26 20:23 EST 2020. Contains 332295 sequences. (Running on oeis4.)