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%I #18 Jun 16 2016 23:27:44
%S 1,3,6,10,17,27,40,56,78,106,140,180,230,290,360,440,535,645,770,910,
%T 1071,1253,1456,1680,1932,2212,2520,2856,3228,3636,4080,4560,5085,
%U 5655,6270,6930,7645,8415,9240,10120,11066,12078
%N Expansion of 1/((1-x)^5*(x^3+x^2+x+1)^2).
%C The Gi2 triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of the Gi2 and other triangle sums see A180662.
%H Harvey P. Dale, <a href="/A189376/b189376.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,2,-6,6,-2,-1,3,-3,1).
%F a(n) = sum(A144678(n-k), k=0..n).
%F Gi2(n) = A189376(n-1) - A189376(n-2) - A189376(n-5) + 2*A189376(n-6) with A189376(n)=0 for n <= -1.
%F a(0)=1, a(1)=3, a(2)=6, a(3)=10, a(4)=17, a(5)=27, a(6)=40, a(7)=56, a(8)=78, a(9)=106, a(10)=140, a(n)=3*a(n-1)-3*a(n-2)+a(n-3)+ 2*a(n-4)- 6*a(n-5)+6*a(n-6)-2*a(n-7)-a(n-8)+3*a(n-9)-3*a(n-10)+a(n-11). - _Harvey P. Dale_, Apr 12 2015
%p a:= n-> coeff (series (1/((1-x)^5*(x^3+x^2+x+1)^2), x, n+1), x, n):
%p seq (a(n), n=0..50);
%t CoefficientList[Series[1/((1-x)^5(x^3+x^2+x+1)^2),{x,0,50}],x] (* or *) LinearRecurrence[{3,-3,1,2,-6,6,-2,-1,3,-3,1},{1,3,6,10,17,27,40,56,78,106,140},50] (* _Harvey P. Dale_, Apr 12 2015 *)
%Y Cf. A139600, A189374, A189375.
%K easy,nonn
%O 0,2
%A _Johannes W. Meijer_, Apr 29 2011
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