cp's OEIS Frontend

From Michael Somos, Mar 04 2004: (Start)

Stirling transform of A005359(n)=[0,2,0,24,0,720,...] is a(n)=[0,2,6,38,270,...].

Stirling transform of -(-1)^n*A052657(n-1)=[0,0,2,-6,48,-240,...] is a(n-1)=[0,0,2,6,38,270,...].

Stirling transform of -(-1)^n*A052558(n-1)=[1,-1,4,-12,72,-360,...] is a(n-1)=[1,0,2,6,38,270,...].

Stirling transform of 2*A052591(n)=[2,4,24,96,...] is a(n+1)=[2,6,38,270,...].

(End)

Also the central moments of a Geometric(1/2) random variable (for example the number of coin tosses until the first head). - Svante Janson, Dec 10 2012

Also the number of ordered set partitions of {1..n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). - Gus Wiseman, Feb 13 2019

Also the number of ordered set partitions of {1..n} with an even number of blocks. - Geoffrey Critzer, Jul 04 2020

Also called the relaxed ménage problem (cf. A000179).

a(n) can be seen as a subset of the unordered pairings of the first 2n integers (A001147) with forbidden pairs (1,2n) and (i,i+1) for all i in [1,2n-1] (all adjacent integers modulo 2n). The linear version of this constraint is A000806. - Olivier Gérard, Feb 08 2011

Number of perfect matchings in the complement of C_{2n} where C_{2n} is the cycle graph on 2n vertices. - Andrew Howroyd, Mar 15 2016

Also the number of 2-uniform set partitions of {1...2n} containing no two cyclically successive vertices in the same block. - Gus Wiseman, Feb 27 2019

See the signed version of these numbers, A000806, for much more information about these numbers.

From Gus Wiseman, Feb 27 2019: (Start)

Also the number of 2-uniform set partitions of {1..2n} containing no two successive vertices in the same block. For example, the a(3) = 5 set partitions are:

{{1,3},{2,5},{4,6}}

{{1,4},{2,5},{3,6}}

{{1,4},{2,6},{3,5}}

{{1,5},{2,4},{3,6}}

{{1,6},{2,4},{3,5}}

(End)

From Gus Wiseman, Jul 05 2020: (Start)

Also the number of permutations of the multiset {1,1,2,2,...,n,n} with no two consecutive terms equal and where the first i appears before the first j for i < j. For example, the a(3) = 5 permutations are the following.

(1,2,3,1,2,3)

(1,2,3,1,3,2)

(1,2,3,2,1,3)

(1,2,3,2,3,1)

(1,2,1,3,2,3)

(End)

Row sums are the Bell numbers (A000110). T(n,0)=A000296(n). T(n,k) = binomial(n,k)*T(n-k,0). Sum(k*T(n,k),k=0..n) = A052889(n) = n*B(n-1), where B(q) are the Bell numbers (A000110).

Exponential Riordan array [exp(exp(x)-1-x),x]. - Paul Barry, Apr 23 2009

Sum_{k=0..n} T(n,k)*2^k = A000110(n+1) is the number of binary relations on an n-set that are both symmetric and transitive. - Geoffrey Critzer, Jul 25 2014

Also the number of set partitions of {1, ..., n} with k cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). Unlike A250104, we count {{1}} as having 1 cyclical adjacency. - Gus Wiseman, Feb 13 2019

This sequence counts set partitions fixed under Callan's conjugation operation.

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. This sequence counts certain self-conjugate set partitions, i.e., fixed points under Callan's conjugation operation.

A cyclical form of A190823.

Also the number of 2-uniform set partitions of {1...2n} such that, when the vertices are arranged uniformly around a circle, no block has its two vertices separated by an arc length of less than 3.

For the definition of a purely crossing partition refer to Dykema link (see PC(n) Definition 1.2 and Table 2).

From Gus Wiseman, Feb 23 2019: (Start)

For n >= 1, a set partition of {1,...,n} is purely crossing if it is topologically connected (A099947), has no successive elements in the same block (A000110(n - 1)), and the first and last vertices belong to different blocks (A005493(n - 2)). For example, the a(4) = 1, a(6) = 5, and a(7) = 14 purely crossing set partitions are:

{{13}{24}} {{135}{246}} {{13}{246}{57}}

{{13}{25}{46}} {{13}{257}{46}}

{{14}{25}{36}} {{135}{26}{47}}

{{14}{26}{35}} {{135}{27}{46}}

{{15}{24}{36}} {{136}{24}{57}}

{{136}{25}{47}}

{{14}{257}{36}}

{{14}{26}{357}}

{{146}{25}{37}}

{{146}{27}{35}}

{{15}{246}{37}}

{{15}{247}{36}}

{{16}{24}{357}}

{{16}{247}{35}}

(End)

A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y.

For the definition of these special purely crossing partitions refer to Dykema link (see PC+(n) Definition 2.1 and Table 2).

From Gus Wiseman, Feb 23 2019: (Start)

a(n) is the number of topologically connected (A099947) set partitions of {1,...,n} with no successive elements in the same block. For example, the a(4) = 1 through a(7) = 19 set partitions are:

{{13}{24}} {{135}{24}} {{135}{246}} {{1357}{246}}

{{13}{25}{46}} {{13}{246}{57}}

{{14}{25}{36}} {{13}{257}{46}}

{{14}{26}{35}} {{135}{26}{47}}

{{15}{24}{36}} {{135}{27}{46}}

{{136}{24}{57}}

{{136}{25}{47}}

{{137}{25}{46}}

{{14}{257}{36}}

{{14}{26}{357}}

{{146}{25}{37}}

{{146}{27}{35}}

{{147}{25}{36}}

{{147}{26}{35}}

{{15}{246}{37}}

{{15}{247}{36}}

{{157}{24}{36}}

{{16}{24}{357}}

{{16}{247}{35}}

(End)