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a(n) = binomial(2*n-1,n) - n^2 - 1.
2

%I #6 Jan 20 2024 16:24:56

%S 0,18,100,425,1666,6370,24228,92277,352594,1351933,5200130,20058103,

%T 77558534,300539938,1166802820,4537567325,17672631538,68923264009,

%U 269128936778,1052049481375,4116715363270,16123801840973,63205303218250,247959266473375,973469712823326

%N a(n) = binomial(2*n-1,n) - n^2 - 1.

%C a(n) is the number of ways to place n indistinguishable balls into n distinguishable boxes with at least 2 boxes remaining empty and not all balls placed in one box.

%t Table[Binomial[2*n - 1, n] - n^2 - 1, {n, 3, 30}] (* _Wesley Ivan Hurt_, Jan 20 2024 *)

%Y Cf. A352027.

%K nonn,easy

%O 3,2

%A _Enrique Navarrete_, Dec 31 2022