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A261724 a(n) is the number of ways of putting n labeled balls into 4 indistinguishable boxes such that each box contains at least 3 balls.

%I A261724 #61 Sep 08 2022 08:46:13
%S A261724 15400,200200,1611610,10335325,57962905,297797500,1439774336,
%T A261724 6662393738,29844199346,130445781284,559533979466,2365296391535,
%U A261724 9885290914059,40944327590760,168389163468240,688631376550260,2803570746766140,11373212443859760,46006062639998890
%N A261724 a(n) is the number of ways of putting n labeled balls into 4 indistinguishable boxes such that each box contains at least 3 balls.
%C A261724 Linear recurrence signature is given by the terms of A255002 after -1. - _Bruno Berselli_, May 20 2016
%H A261724 Colin Barker, <a href="/A261724/b261724.txt">Table of n, a(n) for n = 12..1000</a>
%H A261724 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 (third formula on page 16 is incorrect).
%H A261724 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (30, -415, 3514, -20386, 85924, -272198, 661180, -1244717, 1822478, -2068955, 1802474, -1181760, 563888, -184752, 37152, -3456).
%F A261724 a(n) = (1/12)*(-3^(n - 2)*(n^2 + 5*n + 18) + (1/64)*(2^(2*n + 5) + 3*2^n*(n^4 + 2*n^3 + 19*n^2 + 42*n + 64) - 16*(n^6 - 9*n^5 + 43*n^4 - 91*n^3 + 112*n^2 - 32*n + 8))).
%F A261724 G.f.: x^12*(15400 -261800*x +1996610*x^2 -9045575*x^3 +27162905*x^4 -57079715*x^5 +86268721*x^6 -94696602*x^7 +75062256*x^8 -41952000*x^9 +15705360*x^10 -3538080*x^11 +362880*x^12) / ((1 -x)^7*(1 -2*x)^5*(1 -3*x)^3*(1 -4*x)). - _Colin Barker_, May 24 2016
%t A261724 Table[(1/12) (-(3^(n - 2) (n^2 + 5 n + 18)) + (1/64) (2^(2 n + 5) + 3 2^n (n^4 + 2 n^3 + 19 n^2 + 42 n + 64) - 16 (n^6 - 9 n^5 + 43 n^4 - 91 n^3 + 112 n^2 - 32 n + 8))), {n, 12, 40}]
%o A261724 (Magma) [(1/12)*(-3^(n-2)*(n^2+5*n+18)+(1/64)*(2^(2*n+5)+3*2^n*(n^4+2*n^3+19*n^2+42*n+64)-16*(n^6-9*n^5+43*n^4-91*n^3+112*n^2-32*n+8))): n in [12..40]];
%o A261724 (PARI) Vec(x^12*(15400 -261800*x +1996610*x^2 -9045575*x^3 +27162905*x^4 -57079715*x^5 +86268721*x^6 -94696602*x^7 +75062256*x^8 -41952000*x^9 +15705360*x^10 -3538080*x^11 +362880*x^12) / ((1 -x)^7*(1 -2*x)^5*(1 -3*x)^3*(1 -4*x)) + O(x^30)) \\ _Colin Barker_, May 24 2016
%Y A261724 Cf. A000478, A058844, A272352, A272982, column 4 of A059022.
%K A261724 nonn,easy
%O A261724 12,1
%A A261724 _Vincenzo Librandi_, May 17 2016
%E A261724 Definition, data and formula corrected by _Istvan Mezo_ and _Bruno Berselli_, May 20 2016