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A019677 Expansion of 1/((1-4x)(1-8x)(1-12x)).

%I A019677 #24 Sep 08 2022 08:44:44
%S A019677 1,24,400,5760,77056,989184,12390400,152862720,1867841536,22682271744,
%T A019677 274333696000,3309180026880,39847582498816,479270434504704,
%U A019677 5760041038643200,69190860134154240,830853267268304896,9974742789667160064,119732942204305408000
%N A019677 Expansion of 1/((1-4x)(1-8x)(1-12x)).
%H A019677 Vincenzo Librandi, <a href="/A019677/b019677.txt">Table of n, a(n) for n = 0..200</a>
%H A019677 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (24,-176,384).
%F A019677 a(n) = (4^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).
%F A019677 a(n) = (4^n - 8*8^n + 9*12^n)/2.
%F A019677 G.f.: 1/((1-4*x)*(1-8*x)*(1-12*x)).
%F A019677 E.g.f.: (d^3/dx^3)((((exp(4*x)-1)/4)^3)/3!) = (exp(4*x) - 8*exp(8*x) + 9*exp(12*x))/2.
%F A019677 a(0)=1, a(1)=24, a(2)=400; for n > 2, a(n) = 24*a(n-1) - 176*a(n-2) + 384*a(n-3). - _Vincenzo Librandi_, Jul 03 2013
%F A019677 a(n) = 30*a(n-1) - 96*a(n-2) + 4^n. - _Vincenzo Librandi_, Jul 03 2013
%p A019677 a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [24, -176, 384][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # _Alois P. Heinz_, Jul 03 2013
%t A019677 CoefficientList[Series[1 / ((1 - 4 x) (1 - 8 x) (1 - 12 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jul 03 2013 *)
%t A019677 LinearRecurrence[{24,-176,384},{1,24,400},20] (* _Harvey P. Dale_, Jul 18 2020 *)
%o A019677 (PARI) Vec(1/((1-4*x)*(1-8*x)*(1-12*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o A019677 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-8*x)*(1-12*x)))); /* or */ I:=[1, 24, 400]; [n le 3 select I[n] else 24*Self(n-1)-176*Self(n-2)+384*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 03 2013
%Y A019677 Third column of triangle A075499.
%Y A019677 Cf. A016152, A075907.
%K A019677 nonn,easy
%O A019677 0,2
%A A019677 _N. J. A. Sloane_