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A268579 Expansion of (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).

%I A268579 #30 Feb 26 2016 06:07:57
%S A268579 1,7,11,41,48,120,130,262,275,485,501,807,826,1246,1268,1820,1845,
%T A268579 2547,2575,3445,3476,4532,4566,5826,5863,7345,7385,9107,9150,11130,
%U A268579 11176,13432,13481,16031,16083,18945,19000,22192,22250,25790,25851,29757,29821
%N A268579 Expansion of (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).
%H A268579 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A268579 G.f.: (1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3).
%F A268579 a(n) = Sum_{k = 0..n} (6*k + (-1)^k +3)*(3*k - (-1)^k*(3*k + 1) + 5)/16.
%F A268579 a(n) = 1 + (n*(6*n^2 + 27*n + 35) - (9*n^2 + 15*n + 2)*(-1)^n + 2)/16.
%e A268579 a(0) = 1;
%e A268579 a(1) = 1 + 2*3 = 7;
%e A268579 a(2) = 1 + 2*3 + 4 = 11;
%e A268579 a(3) = 1 + 2*3 + 4 + 5*6 = 41;
%e A268579 a(4) = 1 + 2*3 + 4 + 5*6 + 7 = 48;
%e A268579 a(5) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 = 120;
%e A268579 a(6) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 = 130;
%e A268579 a(7) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12= 262;
%e A268579 a(8) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + 13 = 275;
%e A268579 a(9) = 1 + 2*3 + 4 + 5*6 + 7 + 8*9 + 10 + 11*12 + 13 + 14*15 = 485, etc.
%t A268579 Table[Sum[(6 k + (-1)^k + 3) ((3 k - (-1)^k (3 k + 1) + 5)/16), {k, 0, n}], {n, 0, 42}]
%t A268579 Table[1 + (n (6 n^2 + 27 n + 35) - (9 n^2 + 15 n + 2) (-1)^n + 2)/16, {n, 0, 42}]
%t A268579 LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 7, 11, 41, 48, 120, 130}, 43]
%o A268579 (PARI) Vec((1 + 6*x + x^2 + 12*x^3 - 2*x^4)/((1 - x)^4*(1 + x)^3) + O(x^50)) \\ _Michel Marcus_, Feb 21 2016
%Y A268579 Cf. A000027, A001651, A008585, A016777, A135036, A152743.
%K A268579 nonn,easy
%O A268579 0,2
%A A268579 _Ilya Gutkovskiy_, Feb 21 2016