A188667 Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}.

0, 0, 3, 21, 72, 180, 375, 693, 1176, 1872, 2835, 4125, 5808, 7956, 10647, 13965, 18000, 22848, 28611, 35397, 43320, 52500, 63063, 75141, 88872, 104400, 121875, 141453, 163296, 187572, 214455, 244125, 276768, 312576, 351747, 394485, 441000

Offset: 0

Author

Thomas Wieder, Apr 07 2011

Keywords

• nonn
• easy

Comments

Number of ordered (2,2)-selections which can be taken from the first 2n elements of A008619, the positive integers repeated. Order does count among subselections, e.g. [[1,1],[2,2]] and [[2,2],[1,1]] are different (2,2)-selections. Order does not count within a subselection, e.g. [1,3] is equivalent to [3,1].

Many thanks to Alois P. Heinz, Joerg Arndt, and Olivier Gérard for pointing out bugs in earlier versions of this sequence and for their comments!

The number of (not ordered) (2,2)-selections from natural numbers repeated = A008619 is equal to A086602 (observed by Alois P. Heinz).

The number of ordered (1,1)-selections from natural numbers repeated = A008619 is equal to the squares = A000290.

The number of ordered (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the Oblong numbers = A002378.

The number of ordered (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to A033487.

The number of (not ordered) (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the triangular numbers = A000217.

The number of (not ordered) (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the tritriangular numbers = A050534.

For n>0, the terms of this sequence are related to A014209 by a(n) = sum( i*A014209(i), i=0..n-1 ). [Bruno Berselli, Dec 20 2013]

Examples

			Example: For n=3 there are 21 ordered selections of the type (2,2):
[[1,1],[2,2]], [[1,2],[1,2]], [[2,2],[1,1]], [[1,2],[2,3]],
[[1,3],[2,2]], [[2,2],[1,3]], [[2,3],[1,2]], [[1,1],[2,3]],
[[1,2],[1,3]], [[1,3],[1,2]], [[2,3],[1,1]], [[1,1],[3,3]],
[[1,3],[1,3]], [[3,3],[1,1]], [[1,2],[3,3]], [[1,3],[2,3]],
[[2,3],[1,3]], [[3,3],[1,2]], [[2,2],[3,3]], [[2,3],[2,3]],
[[3,3],[2,2]].
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Quang T. Bach, Roshil Paudyal, Jeffrey B. Remmel, A Fibonacci analogue of Stirling numbers, arXiv:1510.04310 [math.CO], 2015 (see p. 28).
• T. Wieder, Generation of all possible multiselections from a multiset, Progress in Applied Mathematics, 2(1) (2011), 61-66, DOI:10.3968/j.pam.1925252820110201.010. - Thomas Wieder, Oct 15 2011

Crossrefs

Cf. A014209.

Programs

Mathematica

Table[n*(n + 4)*(n - 1)^2/4, {n, 0, 100}] (* Vincenzo Librandi, Oct 18 2012 *)

Formula

a(n) = n*(n+4)*(n-1)^2/4.

G.f.: 3*x^2*(x^2-2*x-1) / (x-1)^5.