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A272352 a(n) is the number of ways of putting n labeled balls into 2 indistinguishable boxes such that each box contains at least 3 balls.

%I A272352 #54 Nov 07 2023 17:41:35
%S A272352 10,35,91,210,456,957,1969,4004,8086,16263,32631,65382,130900,261953,
%T A272352 524077,1048344,2096898,4194027,8388307,16776890,33554080,67108485,
%U A272352 134217321,268435020,536870446,1073741327,2147483119,4294966734,8589933996,17179868553
%N A272352 a(n) is the number of ways of putting n labeled balls into 2 indistinguishable boxes such that each box contains at least 3 balls.
%H A272352 Vincenzo Librandi, <a href="/A272352/b272352.txt">Table of n, a(n) for n = 6..1000</a>
%H A272352 I. Mezo, <a href="http://arxiv.org/abs/1308.1637">Periodicity of the last digits of some combinatorial sequences</a>, arXiv preprint arXiv:1308.1637 [math.CO], 2013 (page 16).
%H A272352 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F A272352 G.f.: x^6*(10 - 15*x + 6*x^2)/((1 - x)^3*(1 - 2*x)).
%F A272352 a(n) = (2^n - 2 - 2*n - 2*binomial(n, 2))/2.
%F A272352 a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), for n > 3.
%F A272352 E.g.f.: (2 - 2*exp(x) + 2*x + x^2)^2/8. - _Stefano Spezia_, Jul 25 2021
%e A272352 For n=6, label the balls A, B, C, D, E, and F. Then each box must contain exactly 3 balls, and the 10 ways are ABC/DEF, ABD/CEF, ABE/CDF, ABF/CDE, ACD/BEF, ACE/BDF, ACF/BDE, ADE/BCF, ADF/BCE, AEF/BCD. - _Michael B. Porter_, Jul 01 2016
%t A272352 Table[1/2 (2^n - 2 - 2 n - 2 Binomial[n, 2]), {n, 6, 40}]
%t A272352 LinearRecurrence[{5,-9,7,-2},{10,35,91,210},30] (* _Harvey P. Dale_, Mar 29 2018 *)
%o A272352 (Magma) [(2^n-2-2*n-2*Binomial(n,2))/2: n in [6..50]];
%Y A272352 Cf. A000478, A058844, A261724, A272982, column 2 of A059022.
%Y A272352 Column k=3 of A201385 (shifted).
%K A272352 nonn,easy
%O A272352 6,1
%A A272352 _Vincenzo Librandi_, May 11 2016