A188667 - OEIS

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

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OFFSET

0,3

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]

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

FORMULA

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

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

EXAMPLE

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]].

MATHEMATICA