A229611 Expansion of 1/((1-x)^3*(1-11x))

1, 14, 160, 1770, 19485, 214356, 2357944, 25937420, 285311665, 3138428370, 34522712136, 379749833574, 4177248169405, 45949729863560, 505447028499280, 5559917313492216, 61159090448414529, 672749994932559990, 7400249944258160080, 81402749386839761090

Offset

0,2

Comments

This sequence was chosen to illustrate a method of matching generating functions and closed-form solutions: The general term associated with the generating function 1/((1-s*x)^3*(1-r*x)) with r>s>=1 is a(n) = [r^(n+3) - s^(n+1)*(s^2 + (r-s)*s*binomial(n+3,1) +(r-s)^2*binomial(n+3,2))] / (r-s)^3 .

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (14,-36,34,-11).

Formula

a(n) = (11^(n+3) - (1 + 10*C(n+3,1) + 100*C(n+3,2)))/1000 = (11^(n+3) - (50*n^2 + 260*n + 331))/1000.

a(n) = 14*a(n-1) -36*a(n-2) +34*a(n-3) -11*a(n-4). - Vincenzo Librandi, Sep 27 2013

Example

a(3) = (11^6 - (50*3^2+260*3 + 331))/1000 = 1770 .

Mathematica

CoefficientList[Series[1/((1 - x)^3 (1 - 11 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 27 2013 *)
LinearRecurrence[{14, -36, 34, -11}, {1, 14, 160, 1770}, 30] (* Harvey P. Dale, Apr 09 2016 *)

Prog

(Magma) [(11^(n+3) - (50*n^2 + 260*n + 331))/1000: n in [0..25]]; // Vincenzo Librandi, Sep 27 2013

Crossrefs

Cf. A002662, A052150, A052161, A052244, A052262.

Sequence in context: A122187 A268946 A343093 * A282043 A193103 A016206

Adjacent sequences: A229608 A229609 A229610 * A229612 A229613 A229614

Keyword

nonn,easy

Author

Yahia Kahloune, Sep 26 2013

Status

approved