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%I #11 Apr 11 2022 22:01:22
%S 0,2,9,30,80,180,357,644,1080,1710,2585,3762,5304,7280,9765,12840,
%T 16592,21114,26505,32870,40320,48972,58949,70380,83400,98150,114777,
%U 133434,154280,177480,203205,231632,262944,297330,334985,376110,420912,469604,522405,579540,641240,707742,779289,856130
%N a(n) = binomial(n,2)*(binomial(n-1,2) + 2).
%C a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with either 1 or 2 boxes remaining empty.
%C a(n) is also the number of weak compositions of n into n parts that contain either one or two 0's.
%C a(n)+1 is the number of ways to place n indistinguishable balls into n distinguishable boxes with at most 2 boxes remaining empty (just add the case of no empty boxes in which we place exactly one ball in one box).
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: x^2*(2 - x + 5*x^2)/(1 - x)^5. - _Stefano Spezia_, Mar 15 2022
%e a(4)=30 since 4 can be written as 3+1+0+0, 0+3+0+1, etc. (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions).
%t a[n_] := Binomial[n, 2] * (Binomial[n - 1, 2] + 2); Array[a, 50] (* _Amiram Eldar_, Mar 15 2022 *)
%Y Cf. A010763, A350653.
%K nonn,easy
%O 1,2
%A _Enrique Navarrete_, Mar 14 2022