A145132 Expansion of x/((1 - x - x^4)*(1 - x)^3).

0, 1, 4, 10, 20, 36, 61, 99, 155, 236, 352, 517, 750, 1077, 1534, 2171, 3057, 4287, 5992, 8353, 11620, 16138, 22383, 31012, 42932, 59395, 82129, 113519, 156857, 216687, 299281, 413296, 570681, 787929, 1087805, 1501731, 2073078, 2861710, 3950256, 5452767

Offset: 0

Author

Alois P. Heinz, Oct 03 2008

Keywords

• nonn
• easy

Comments

The coefficients of the recursion for a(n) are given by the 4th row of A145152.

Examples

			a(8) = 155 = 4*99 -6*61 +4*36 -3*10 +3*4 -1.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,0,-3,3,-1).

Crossrefs

4th column of A145153. Cf. A145152.

Programs

Maple

col:= proc(k) local l, j, M, n; l:= `if`(k=0, [1, 0, 0, 1], [seq(coeff( -(1-x-x^4) *(1-x)^(k-1), x, j), j=1..k+3)]); M:= Matrix(nops(l), (i,j)-> if i=j-1 then 1 elif j=1 then l[i] else 0 fi); `if`(k=0, n->(M^n)[2,3], n->(M^n)[1,2]) end: a:= col(4): seq(a(n), n=0..40);

Mathematica

CoefficientList[Series[x / ((1 - x - x^4) (1 - x)^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 06 2013 *)
LinearRecurrence[{4,-6,4,0,-3,3,-1},{0,1,4,10,20,36,61},40] (* Harvey P. Dale, Apr 04 2014 *)

PARI

concat(0,Vec(1/(1-x-x^4)/(1-x)^3+O(x^99))) \\ Charles R Greathouse IV, Sep 25 2012

Formula

a(n) = 4a(n-1) -6a(n-2) +4a(n-3) -3a(n-5) +3a(n-6) -a(n-7).