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A229834 Expansion of (1+4*x+x^2) / ((1-x)^3*(1+x)^4).

%I A229834 #38 Jan 27 2022 13:00:25
%S A229834 1,3,1,11,-2,26,-10,50,-25,85,-49,133,-84,196,-132,276,-195,375,-275,
%T A229834 495,-374,638,-494,806,-637,1001,-805,1225,-1000,1480,-1224,1768,
%U A229834 -1479,2091,-1767,2451,-2090,2850,-2450,3290,-2849,3773,-3289,4301,-3772,4876,-4300,5500,-4875,6175,-5499,6903,-6174,7686,-6902
%N A229834 Expansion of (1+4*x+x^2) / ((1-x)^3*(1+x)^4).
%C A229834 The sequence can be generated in the following way:
%C A229834 ---------------------------       --------------------------
%C A229834       [0]  [1] [2] [3]  [4]  ...             [i]
%C A229834 ---------------------------       --------------------------
%C A229834 [0]    1,   1,  1,  1,   1,  ...  t(0,i) =    1
%C A229834 [1]    7,   6,  5,  4,   3,  ...  t(1,i) = t(1,i-1) - t(0,i)
%C A229834 [2]   19,  13,  8,  4,   1,  ...  t(2,i) = t(2,i-1) - t(1,i)
%C A229834 [3]   37,  24, 16, 12,  11,  ...  t(3,i) = t(3,i-1) - t(2,i)
%C A229834 [4]   61,  37, 21,  9,  -2,  ...  t(4,i) = t(4,i-1) - t(3,i)
%C A229834 [5]   91,  54, 33, 24,  26,  ...             etc.
%C A229834 [6]  127,  73, 40, 16, -10,  ...
%C A229834 [7]  169,  96, 56, 40,  50,  ...
%C A229834 [8]  217, 121, 65, 25, -25,  ...
%C A229834 [9]  271, 150, 85, 60,  85,  ...
%C A229834 ...
%C A229834 Column 0 is A003215;
%C A229834 column 1 is A032528;
%C A229834 column 2 is A001082;
%C A229834 column 3 is A241496;
%C A229834 column 4 is this sequence.
%C A229834 The third differences are 16, -35, 64, -105, 160, ..., a signed variant of A077415. - _R. J. Mathar_, Apr 18 2014
%H A229834 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,3,-3,-3,1,1).
%F A229834 G.f.: (1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^4). - _R. J. Mathar_, Apr 18 2014
%F A229834 a(n) = a(-n-5) = 1 + n*(n + 5)*(9 - (2*n + 5)*(-1)^n)/48. [_Bruno Berselli_, Apr 22 2014]
%t A229834 Table[1 + n (n + 5) (9 - (2 n + 5) (-1)^n)/48, {n, 0, 60}] (* _Bruno Berselli_, Apr 22 2014 *)
%t A229834 CoefficientList[Series[(1+4x+x^2)/((1-x)^3(1+x)^4),{x,0,60}],x] (* or *) LinearRecurrence[{-1,3,3,-3,-3,1,1},{1,3,1,11,-2,26,-10},60] (* _Harvey P. Dale_, Jan 27 2022 *)
%Y A229834 Cf. A077415; A058373: a(2k) = -A058373(k); A051925: a(2k+1) = A051925(k+2).
%Y A229834 Columns of the table in Comments section: A001082, A003215, A032528.
%K A229834 sign,easy
%O A229834 0,2
%A A229834 _Stefano Maruelli_, Dec 19 2013