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A229834
Expansion of (1+4*x+x^2) / ((1-x)^3*(1+x)^4).
1
1, 3, 1, 11, -2, 26, -10, 50, -25, 85, -49, 133, -84, 196, -132, 276, -195, 375, -275, 495, -374, 638, -494, 806, -637, 1001, -805, 1225, -1000, 1480, -1224, 1768, -1479, 2091, -1767, 2451, -2090, 2850, -2450, 3290, -2849, 3773, -3289, 4301, -3772, 4876, -4300, 5500, -4875, 6175, -5499, 6903, -6174, 7686, -6902
OFFSET
0,2
COMMENTS
The sequence can be generated in the following way:
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[0] [1] [2] [3] [4] ... [i]
--------------------------- --------------------------
[0] 1, 1, 1, 1, 1, ... t(0,i) = 1
[1] 7, 6, 5, 4, 3, ... t(1,i) = t(1,i-1) - t(0,i)
[2] 19, 13, 8, 4, 1, ... t(2,i) = t(2,i-1) - t(1,i)
[3] 37, 24, 16, 12, 11, ... t(3,i) = t(3,i-1) - t(2,i)
[4] 61, 37, 21, 9, -2, ... t(4,i) = t(4,i-1) - t(3,i)
[5] 91, 54, 33, 24, 26, ... etc.
[6] 127, 73, 40, 16, -10, ...
[7] 169, 96, 56, 40, 50, ...
[8] 217, 121, 65, 25, -25, ...
[9] 271, 150, 85, 60, 85, ...
...
Column 0 is A003215;
column 1 is A032528;
column 2 is A001082;
column 3 is A241496;
column 4 is this sequence.
The third differences are 16, -35, 64, -105, 160, ..., a signed variant of A077415. - R. J. Mathar, Apr 18 2014
FORMULA
G.f.: (1 + 4*x + x^2) / ((1 - x)^3*(1 + x)^4). - R. J. Mathar, Apr 18 2014
a(n) = a(-n-5) = 1 + n*(n + 5)*(9 - (2*n + 5)*(-1)^n)/48. [Bruno Berselli, Apr 22 2014]
MATHEMATICA
Table[1 + n (n + 5) (9 - (2 n + 5) (-1)^n)/48, {n, 0, 60}] (* Bruno Berselli, Apr 22 2014 *)
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 *)
CROSSREFS
Cf. A077415; A058373: a(2k) = -A058373(k); A051925: a(2k+1) = A051925(k+2).
Columns of the table in Comments section: A001082, A003215, A032528.
Sequence in context: A230262 A323854 A256589 * A120291 A099001 A119947
KEYWORD
sign,easy
AUTHOR
Stefano Maruelli, Dec 19 2013
STATUS
approved