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A108368 Coefficients of x/(1-3*x-3*x^2-x^3).

%I A108368 #49 May 29 2025 16:30:34
%S A108368 0,1,3,12,46,177,681,2620,10080,38781,149203,574032,2208486,8496757,
%T A108368 32689761,125768040,483870160,1861604361,7162191603,27555258052,
%U A108368 106013953326,407869825737,1569206595241,6037243216260,23227219260240,89362594024741,343806683071203
%N A108368 Coefficients of x/(1-3*x-3*x^2-x^3).
%C A108368 From _Enrique Navarrete_, Jul 09 2024: (Start)
%C A108368 a(n+1) is the number of generalized compositions of n using parts of size at most 3 where there are binomial(3,i) types of i.
%C A108368 For example, the following table gives the type of composition, the number of such compositions, and the total number of compositions of n = 5 using parts of size at most 3 where there are binomial(3,i) types of i (ie. 3 types of 1, 3 types of 2 and 1 type of 3):
%C A108368     Type                     Number              Total
%C A108368     3+2                        2                    6
%C A108368     3+1+1                      3                   27
%C A108368     2+2+1                      3                   81
%C A108368     2+1+1+1                    4                  324
%C A108368     1+1+1+1+1                  1                  243,
%C A108368 adding to a(6) = 681.
%C A108368 The coefficients of 1/(1 - C(k,1)*x - C(k,2)*x^2 - C(k,3)*x^3 - ... - C(k,k)*x^k) give the number of generalized compositions of n using parts of size at most k where there are binomial(k,i) types of i. (End).
%C A108368 For n>0, the expansion of (4^(1/3) + 2^(1/3) + 1)^n is a(n)*4^(1/3) + (a(n) + a(n-1))*2^(1/3) + (a(n) + 2*a(n-1) + a(n-2)). - _Greg Dresden_, Aug 14 2024
%D A108368 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 562.
%H A108368 Michael De Vlieger, <a href="/A108368/b108368.txt">Table of n, a(n) for n = 0..1710</a>
%H A108368 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 4.
%H A108368 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,1).
%F A108368 x=a(n), z=a(-n), y=a(n)+a(n-1), t=a(-n)+a(-n-1) is a solution to 2(x^3+z^3)=y^3+t^3.
%F A108368 G.f.: x/(1-3*x-3*x^2-x^3).
%F A108368 a(n) = 3*a(n-1)+3*a(n-2)+a(n-3).
%F A108368 a(-1-n) = A108369(n).
%F A108368 a(n+1) = Sum_{k>=0} (1/2)^(k+1) * binomial(3*k,n). - _Seiichi Manyama_, Aug 03 2024
%t A108368 CoefficientList[Series[x/(1-3*x-3*x^2-x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,3,1},{0,1,3},40] (* _Vladimir Joseph Stephan Orlovsky_, Jan 31 2012 *)
%o A108368 (PARI) a(n)=if(n>=0, polcoeff(x/(1-3*x-3*x^2-x^3)+x*O(x^n),n), n=-1-n; polcoeff(x/(1+3*x+3*x^2-x^3)+x*O(x^n),n))
%Y A108368 Cf. A000012, A000129, A374454, A374455.
%Y A108368 Cf. A108369.
%K A108368 nonn,easy
%O A108368 0,3
%A A108368 _Michael Somos_, Jun 01 2005