A021514 Expansion of 1/((1-x)(1-3x)(1-6x)(1-10x)).

1, 20, 273, 3208, 35069, 368988, 3800761, 38676176, 390782997, 3931986916, 39464899409, 395519441304, 3960417893485, 39635522209004, 396543288909417, 3966561311533792, 39672383714545733, 396764460934414452, 3967888352659017985, 39680345988222812840

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (20,-127,288,-180).

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-6*x)*(1-10*x)))); // Vincenzo Librandi, Jul 10 2013

Magma

I:=[1, 20, 273, 3208]; [n le 4 select I[n] else 20*Self(n-1)-127*Self(n-2)+288*Self(n-3)-180*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Jul 10 2013

Mathematica

CoefficientList[Series[1 / ((1 - x) (1 - 3 x) (1 - 6 x) (1 - 10 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 10 2013 *)
LinearRecurrence[{20,-127,288,-180},{1,20,273,3208},20] (* Harvey P. Dale, Feb 13 2022 *)

PARI

Vec(1/((1-x)*(1-3*x)*(1-6*x)*(1-10*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (10^(n+4) - 7*6^(n+4) + 20*3^(n+4) - 28)/2520. [Yahia Kahloune, Jun 19 2013]

a(0)=1, a(1)=20; for n>1, a(n) = 16*(n-1) -60*a(n-2) +(3^n -1)/2. - Vincenzo Librandi, Jul 10 2013

a(0)=1, a(1)=20, a(2)=273, a(3)=3208; for n>3, a(n) = 20*a(n-1) -127*a(n-2) +288*a(n-3) -180*a(n-4). - Vincenzo Librandi, Jul 10 2013