A272352 - OEIS

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 #54 Nov 07 2023 17:41:35

%S 10,35,91,210,456,957,1969,4004,8086,16263,32631,65382,130900,261953,

%T 524077,1048344,2096898,4194027,8388307,16776890,33554080,67108485,

%U 134217321,268435020,536870446,1073741327,2147483119,4294966734,8589933996,17179868553

%N 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 Vincenzo Librandi, <a href="/A272352/b272352.txt">Table of n, a(n) for n = 6..1000</a>

%H 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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).

%F G.f.: x^6*(10 - 15*x + 6*x^2)/((1 - x)^3*(1 - 2*x)).

%F a(n) = (2^n - 2 - 2*n - 2*binomial(n, 2))/2.

%F a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4), for n > 3.

%F E.g.f.: (2 - 2*exp(x) + 2*x + x^2)^2/8. - _Stefano Spezia_, Jul 25 2021

%e 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 Table[1/2 (2^n - 2 - 2 n - 2 Binomial[n, 2]), {n, 6, 40}]

%t LinearRecurrence[{5,-9,7,-2},{10,35,91,210},30] (* _Harvey P. Dale_, Mar 29 2018 *)

%o (Magma) [(2^n-2-2*n-2*Binomial(n,2))/2: n in [6..50]];

%Y Cf. A000478, A058844, A261724, A272982, column 2 of A059022.

%Y Column k=3 of A201385 (shifted).

%K nonn,easy

%O 6,1

%A _Vincenzo Librandi_, May 11 2016