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Expansion of 1/((1-8x)(1-9x)(1-10x)(1-12x)).
1

%I #30 Aug 15 2022 04:26:59

%S 1,39,955,18795,325171,5167659,77353795,1107933915,15345293491,

%T 207057429579,2736619356835,35572444360635,456191563128211,

%U 5785999811488299,72719715037363075,907085298655532955

%N Expansion of 1/((1-8x)(1-9x)(1-10x)(1-12x)).

%H Vincenzo Librandi, <a href="/A016091/b016091.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 (39,-566,3624,-8640).

%F From _Vincenzo Librandi_, Mar 17 2011: (Start)

%F a(n) = 39*a(n-1) - 566*a(n-2) + 3624*a(n-3) - 8640*a(n-4) for n >= 4.

%F a(n) = 22*a(n-1) - 120*a(n-2) + 9^(n+1) - 8^(n+1) for n >= 2. (End)

%F a(n) = -8^(n+2) + 3*9^(n+2) - 25*10^(n+1) + 6*12^(n+1). - _R. J. Mathar_, Mar 18 2011

%F E.g.f.: exp(8*x)*(72*exp(4*x) - 250*exp(2*x) + 243*exp(x) - 64). - _Stefano Spezia_, Aug 14 2022

%t CoefficientList[Series[1 / ((1 - 8 x) (1 - 9 x) (1 - 10 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)

%t LinearRecurrence[{39,-566,3624,-8640},{1,39,955,18795},30] (* _Harvey P. Dale_, Jul 08 2019 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-8*x)*(1-9*x)*(1-10*x)*(1-12*x)))); /* or */ I:=[1, 39, 955, 18795]; [n le 4 select I[n] else 39*Self(n-1)-566*Self(n-2)+3624*Self(n-3)-8640*Self(n-4): n in [1..20]]; // _Vincenzo Librandi_, Jun 24 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_