%I #28 Mar 09 2026 09:53:13

%S 1,25,427,6229,83779,1076341,13459699,165601573,2017494787,

%T 24431946517,294797887891,3549239159557,42674698231075,

%U 512696237681653,6156632228705203,73909998565124581,887135686636037443,10647155093273691349,127776650654729658835,1533395943964613455045

%N Expansion of g.f. 1/((1 - 6*x)*(1 - 7*x)*(1 - 12*x)).

%H Vincenzo Librandi, <a href="/A020577/b020577.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (25,-198,504).

%F a(n) = 6*6^n -49*7^n/5 +24*12^n/5. - _R. J. Mathar_, Jun 30 2013

%F a(0)=1, a(1)=25, a(2)=427; for n>2, a(n) = 25*a(n-1) -198*a(n-2) +504*a(n-3). - _Vincenzo Librandi_, Jul 04 2013

%F a(n) = 19*a(n-1) -84*a(n-2) +6^n. - _Vincenzo Librandi_, Jul 04 2013

%F E.g.f.: exp(6*x)*(30 - 49*exp(x) + 24*exp(6*x))/5. - _Stefano Spezia_, Mar 09 2026

%t CoefficientList[Series[1 / ((1 - 6 x) (1 - 7 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 04 2013 *)

%t LinearRecurrence[{25,-198,504},{1,25,427},30] (* _Harvey P. Dale_, Sep 27 2014 *)

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-7*x)*(1-12*x)))); // _Vincenzo Librandi_, Jul 04 2013

%o (Magma) I:=[1, 25, 427]; [n le 3 select I[n] else 25*Self(n-1)-198*Self(n-2)+504*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 04 2013

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_