%I #22 May 01 2025 08:31:22

%S 1,15,160,1530,14071,127605,1151470,10372560,93381541,840520395,

%T 7564945180,68085295590,612770035411,5514937460385,49634458601290,

%U 446710191850620,4020391920103681,36183527861539575,325651752496199800

%N Expansion of 1/((1-x)*(1-2x)*(1-3x)*(1-9x)).

%H Vincenzo Librandi, <a href="/A021048/b021048.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (15,-65,105,-54).

%F a(n) = (243/112)*9^n - (9/4)*3^n + (8/7)*2^n - 1/16. - Neven Juric, Oct 22 2009

%F a(0)=1, a(1)=15; for n>1, a(n) = 12*a(n-1) -27*a(n-2) +2^n - 1. - _Vincenzo Librandi_, Jul 05 2013

%F a(0)=1, a(1)=15, a(2)=160, a(3)=1530; for n>3, a(n) = 15*a(n-1) -65*a(n-2) +105*a(n-3) -54*a(n-4). - _Vincenzo Librandi_, Jul 05 2013

%F E.g.f.: exp(x)*(243*exp(8*x) - 252*exp(2*x) + 128*exp(x) - 7)/112. - _Stefano Spezia_, Apr 30 2025

%t CoefficientList[Series[1 / ((1 - x) (1 - 2 x) (1 - 3 x) (1 - 9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 05 2013 *)

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-2*x)*(1-3*x)*(1-9*x)))); // _Vincenzo Librandi_, Jul 05 2013

%o (Magma) I:=[1, 15, 160, 1530]; [n le 4 select I[n] else 15*Self(n-1)-65*Self(n-2)+105*Self(n-3)-54*Self(n-4): n in [1..25]]; // _Vincenzo Librandi_, Jul 05 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_