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%I #39 Sep 08 2022 08:46:16
%S 280,2100,10395,42735,158301,549549,1827826,5903898,18682014,58257810,
%T 179765973,550478241,1676305723,5083927299,15372843684,46383762084,
%U 139730030100,420448298400,1264071094975,3798101973315,11406989362185,34248214131465,102803026929030,308533903071390
%N a(n) is the number of ways of putting n labeled balls into 3 indistinguishable boxes such that each box contains at least 3 balls.
%H Vincenzo Librandi, <a href="/A272982/b272982.txt">Table of n, a(n) for n = 9..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 (second formula on page 16 is incorrect).
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (14,-85,294,-639,906,-839,490,-164,24).
%F G.f.: x^9*(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5).
%F a(n) = (1/3)*(1/16)*(6*n^4 - 12*n^3 - 3*2^n*n^2 + 42*n^2 - 9*2^n*n + 12*n + 8*3^n - 3*2^(n+3) + 24).
%F a(n) = 3*a(n-1) + C(n-1,2)*(2^(n-4) + 2 - n - C(n-3, 2)), a(n)=0, n < 9. - _Vladimir Kruchinin_, Oct 04 2018
%e For n=9, label the balls A through I. The box containing ball A can contain 8*7/2 = 28 combinations of other balls. There are 6 balls for the other two boxes, so there are A272352(6) = 10 combinations for those two boxes. Thus, a(9) = 28*10 = 280. - _Michael B. Porter_, Jul 01 2016
%t Table[(1/3) (1/16) (6 n^4 - 12 n^3 - 3 2^n n^2 + 42 n^2 - 9 2^n n + 12 n + 8 3^n - 3 2^(n + 3) + 24), {n, 9, 40}]
%t CoefficientList[Series[(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5), {x, 0, 40}], x] (* _Stefano Spezia_, Oct 04 2018 *)
%o (Magma) [(1/3)*(1/16)*(6*n^4-12*n^3-3*2^n*n^2+42*n^2-9*2^n*n+12*n+8*3^n-3*2^(n+3)+24): n in [9..40]];
%o (PARI) Vec(x^9*(280 - 1820*x + 4795*x^2 - 6615*x^3 + 5106*x^4 - 2100*x^5 + 360*x^6)/((1 - 3*x)*(1 - 2*x)^3*(1 - x)^5) + O(x^40)) \\ _Stefano Spezia_, Oct 04 2018
%Y Cf. A000478, A058844, A261724, A272352, column 3 of A059022.
%K nonn,easy
%O 9,1
%A _Vincenzo Librandi_, May 12 2016
%E Data, formulas and programs corrected for erroneous formula in Mezo's paper by _Bruno Berselli_, May 21 2016