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A189374 Expansion of 1/((1-x)^5*(x^2+x+1)^3).

%I A189374 #16 Jul 28 2016 13:39:09
%S A189374 1,2,3,7,11,15,25,35,45,65,85,105,140,175,210,266,322,378,462,546,630,
%T A189374 750,870,990,1155,1320,1485,1705,1925,2145,2431,2717,3003,3367,3731,
%U A189374 4095,4550,5005,5460,6020,6580,7140,7820
%N A189374 Expansion of 1/((1-x)^5*(x^2+x+1)^3).
%C A189374 The Ca1(n) and Ze3(n) triangle sums of A139600 lead to the sequence given above, see the formulas. For the definitions of these triangle sums see A180662.
%H A189374 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,3,-6,3,-3,6,-3,1,-2,1)
%F A189374 a(n) = (2*a(n-1) + 2*a(n-2) + (8+n)*a(n-3))/n with a(0)=1, a(1)=2, a(2)=3 and a(3)=7.
%F A189374 a(n) = sum(A011779(n-k)*A049347(k), k=0..n).
%F A189374 Ca1(n) = A189374(n-3) - A189374(n-4) - A189374(n-6) + 2*A189374(n-7).
%F A189374 Ze3(n) = 2*A189374(n-3) - A189374(n-4) - 2*A189374(n-6) + 5*A189374(n-7) with A189374(n)=0 for n <= -1.
%F A189374 a(n) = (floor(n/3)+1)*(floor(n/3)+2)*(floor(n/3)+3)*(3*floor(n/3)+4*(4-(3*floor((n+3)/3)-n)))/24. - _Luce ETIENNE_, Jun 29 2015
%p A189374 a:= proc(n) option remember; `if` (n<4, [1, 2, 3, 7][n+1], (2*a(n-1) +2*a(n-2) +(8+n) *a(n-3))/n) end: seq (a(n), n=0..50);
%Y A189374 Cf. A139600, A189375, A189376.
%K A189374 easy,nonn
%O A189374 0,2
%A A189374 _Johannes W. Meijer_, Apr 29 2011