%I #41 Oct 28 2025 04:00:52

%S 0,0,3,21,72,180,375,693,1176,1872,2835,4125,5808,7956,10647,13965,

%T 18000,22848,28611,35397,43320,52500,63063,75141,88872,104400,121875,

%U 141453,163296,187572,214455,244125,276768,312576,351747,394485,441000,491508,546231,605397

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

%C Number of ordered (2,2)-selections which can be taken from the first 2*n 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].

%C 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!

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

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

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

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

%C 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.

%C 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.

%C For n>0, the terms of this sequence are related to A014209 by a(n) = Sum_{i=0..n-1} i*A014209(i). - _Bruno Berselli_, Dec 20 2013

%H Vincenzo Librandi, <a href="/A188667/b188667.txt">Table of n, a(n) for n = 0..1000</a>

%H Quang T. Bach, Roshil Paudyal, and Jeffrey B. Remmel, <a href="https://arxiv.org/abs/1510.04310">A Fibonacci analogue of Stirling numbers</a>, arXiv:1510.04310 [math.CO], 2015 (see p. 28).

%H Thomas Wieder, <a href="https://doi.org/10.3968/j.pam.1925252820110201.010">Generation of all possible multiselections from a multiset</a>, Progress in Applied Mathematics, 2(1) (2011), 61-66, DOI:10.3968/j.pam.1925252820110201.010.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

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

%F a(n) = 3 * A014205(n-1) for n>=1. - _Alois P. Heinz_, Oct 27 2025

%F E.g.f.: exp(x)*x^2*(6 + 8*x + x^2)/4. - _Stefano Spezia_, Oct 28 2025

%e For n=3 there are 21 ordered selections of the type (2,2):

%e [[1,1],[2,2]], [[1,2],[1,2]], [[2,2],[1,1]], [[1,2],[2,3]],

%e [[1,3],[2,2]], [[2,2],[1,3]], [[2,3],[1,2]], [[1,1],[2,3]],

%e [[1,2],[1,3]], [[1,3],[1,2]], [[2,3],[1,1]], [[1,1],[3,3]],

%e [[1,3],[1,3]], [[3,3],[1,1]], [[1,2],[3,3]], [[1,3],[2,3]],

%e [[2,3],[1,3]], [[3,3],[1,2]], [[2,2],[3,3]], [[2,3],[2,3]],

%e [[3,3],[2,2]].

%t Table[n*(n + 4)*(n - 1)^2/4, {n, 0, 100}] (* _Vincenzo Librandi_, Oct 18 2012 *)

%Y Cf. A014205, A014209.

%K nonn,easy

%O 0,3

%A _Thomas Wieder_, Apr 07 2011