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A301922 Regular triangle where T(n,k) is the number of unlabeled k-uniform hypergraphs spanning n vertices. 12
1, 1, 1, 1, 2, 1, 1, 7, 3, 1, 1, 23, 29, 4, 1, 1, 122, 2102, 150, 5, 1, 1, 888, 7011184, 7013164, 1037, 6, 1, 1, 11302, 1788775603336, 29281354507753848, 1788782615612, 12338, 7, 1, 1, 262322, 53304526022885280592, 234431745534048893449761040648512, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..91

FORMULA

T(n,k) = A309858(n,k) - A309858(n-1,k). - Alois P. Heinz, Aug 21 2019

EXAMPLE

Triangle begins:

   1

   1   1

   1   2   1

   1   7   3   1

   1  23  29   4   1

The T(4,2) = 7 hypergraphs:

  {{1,2},{3,4}}

  {{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}}

MAPLE

g:= (l, i, n)-> `if`(i=0, `if`(n=0, [[]], []), [seq(map(x->

     [x[], j], g(l, i-1, n-j))[], j=0..min(l[i], n))]):

h:= (p, v)-> (q-> add((s-> add(`if`(andmap(i-> irem(k[i], p[i]

     /igcd(t, p[i]))=0, [$1..q]), mul((m-> binomial(m, k[i]*m

     /p[i]))(igcd(t, p[i])), i=1..q), 0), t=1..s)/s)(ilcm(seq(

    `if`(k[i]=0, 1, p[i]), i=1..q))), k=g(p, q, v)))(nops(p)):

b:= (n, i, l, v)-> `if`(n=0 or i=1, 2^((p-> h(p, v))([l[], 1$n]))

     /n!, add(b(n-i*j, i-1, [l[], i$j], v)/j!/i^j, j=0..n/i)):

A:= proc(n, k) A(n, k):= `if`(k>n-k, A(n, n-k), b(n$2, [], k)) end:

T:= (n, k)-> A(n, k)-A(n-1, k):

seq(seq(T(n, k), k=1..n), n=1..9);  # Alois P. Heinz, Aug 21 2019

PROG

(PARI)

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(#L<k, listput(L, #L))); Vec(L)}

can(v, f)={my(d=1, u=v); while(d>0, 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!}

for(n=1, 10, for(k=1, n, print1(U(n, k)-U(n-1, k), ", ")); print) \\ Andrew Howroyd, Aug 10 2019

CROSSREFS

Row sums are A301481. Second column is A002494.

Cf. A003465, A055621, A298422, A298426, A299471, A301481, A301920, A306017-A306021, A309858.

Sequence in context: A178234 A259175 A297431 * A144510 A143670 A169730

Adjacent sequences:  A301919 A301920 A301921 * A301923 A301924 A301925

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Jun 19 2018

EXTENSIONS

Terms a(16) and beyond from Andrew Howroyd, Aug 09 2019

STATUS

approved

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Last modified October 16 13:19 EDT 2019. Contains 328082 sequences. (Running on oeis4.)