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A188667 Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}. 1
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 (list; graph; refs; listen; history; text; internal format)
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 GĂ©rard 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

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

CROSSREFS

Cf. A014209.

Sequence in context: A117984 A050615 A145658 * A083564 A281008 A238193

Adjacent sequences:  A188664 A188665 A188666 * A188668 A188669 A188670

KEYWORD

nonn,easy

AUTHOR

Thomas Wieder, Apr 07 2011

STATUS

approved

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Last modified October 18 06:12 EDT 2018. Contains 316304 sequences. (Running on oeis4.)