A122836 -id:A122836 - cp's OEIS frontend

A178689 Partial sums of A122836 (number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology).

0, 0, 0, 10, 253, 6384, 208887, 9673189, 651633791, 63901292323, 9040801794022, 1825884406581355, 521181413335003984, 208402574279716434454, 115825454535371969786250, 88852094572776191675804592

Offset: 0

Jonathan Vos Post, Dec 25 2010

nonn

All listed (the first 16) values are nonprimes. In the underlying sequence, only A122836(5) = 6131 is prime of the listed values.

			a(4) = 0 + 0 + 0 + 10 + 243 = 253 = 11 * 23.

Cf. A000798, A122835, A122836.

a(n) = Sum_{i=0..n} A122836(i) = Sum_{i=0..n} (A000798(i) - A122835(i)) = (Sum_{i=0..n} A000798(i)) - (Sum_{i=0..n} A122835(i)).

a(13)-a(15) corrected by Georg Fischer, Dec 26 2022

A122835 Number of topologies on n labeled elements in which no element belongs to any pair of noncomparable members of the topology.

Original entry on oeis.org

1, 1, 4, 19, 112, 811, 7024, 70939, 818752, 10630891, 153371344, 2433948859, 42137351392, 790287522571, 15962014455664, 345424786466779, 7973482022972032, 195556150543703851, 5078301994885267984

Offset: 0

Nathan K. McGregor (mcgregnk(AT)ese.wustl.edu), Sep 15 2006

The number of topologies on n labeled elements is a fundamental sequence (A000798), which many mathematicians believe is impossible to completely determine.
The present sequence is an elegant recursion that enumerates the topologies on n labeled elements that can be "drawn" (as, for example, on page 76 of Munkres) in such a way that the boundaries of the subsets do not "cross" one another. Thus I recommend that topologies be classified as "planar" if their members can be drawn without crossings and "non-planar" otherwise.
This is analogous to the way in which subgroup lattices are called planar or non-planar. Using this terminology, the above sequence gives the number of planar topologies on n labeled elements. If the number of non-planar topologies on n labeled elements (see A122836) could be enumerated, then so could the total number of topologies on n labeled elements.
Another way to state the definition is that any two members of the topology are comparable or disjoint. - Rainer Rosenthal, Jan 02 2011
Conjectural closed form for n>0: 3*2^(k-3)(LerchPhi[1/4, -k, 1/2] + 2 PolyLog[-k, 1/4]) - 1/2. - Vladimir Reshetnikov, Jan 07 2011

J. Munkres, Topology, Prentice Hall, (2000), p. 76.

Cf. A000798, A122836.

a122835:=proc(n) option remember; if n=0 then 1 else 2^(n-1) - 1 + add(a122835(n-k)*binomial(n,k),k=1..n); fi; end;

a[n_]:=a[n]=2^(n-1)-1+Sum[a[n-k]*Binomial[n,k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,25}]
a[ n_] := (3/4) * (PolyLog[ -n, 1/2] + Boole[n==0]) - 1/2 (* Michael Somos, Jan 07 2011 *)

{a(n) = local(A); if( n<1, n==0, A = exp(x + x * O(x^n)) / 2; n! * polcoeff( (3/4) / (1 - A) - A, n))} /* Michael Somos, Jan 07 2011 */

a(n) = 2^(n-1) - 1 + Sum{C(n,k)*a(n-k), k = 1 ... n}
E.g.f.: (3/4) / (1 - exp(x)/2) - exp(x)/2. - Michael Somos, Jan 07 2011
a(n) = (A000629(n) + 0^n) * (3/4) - 1/2. - Michael Somos, Jan 07 2011

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A178689 Partial sums of A122836 (number of topologies on n labeled elements in which at least one element belongs to some pair of noncomparable members of the topology).

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A122835 Number of topologies on n labeled elements in which no element belongs to any pair of noncomparable members of the topology.

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