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A020570 Expansion of g.f. 1/((1-6*x)*(1-7*x)*(1-8*x)).

%I A020570 #29 Mar 26 2025 17:33:37
%S A020570 1,21,295,3465,36751,365001,3463615,31794105,284628751,2499039081,
%T A020570 21606842335,184519243545,1559982264751,13079717026761,
%U A020570 108915112739455,901732722577785,7429565635164751,60963378722560041,498496565225842975,4064108629664292825,33049477950757248751
%N A020570 Expansion of g.f. 1/((1-6*x)*(1-7*x)*(1-8*x)).
%H A020570 Vincenzo Librandi, <a href="/A020570/b020570.txt">Table of n, a(n) for n = 0..200</a>
%H A020570 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-146,336).
%F A020570 If we define f(m,j,x) = Sum_{k=j..m} (binomial(m,k)*Stirling2(k,j)*x^(m-k)) then a(n-2) = f(n,2,6), (n>=2). - _Milan Janjic_, Apr 26 2009
%F A020570 a(n) = 18*6^n - 49*7^n + 32*8^n. - _R. J. Mathar_, Jun 30 2013
%F A020570 From _Vincenzo Librandi_, Jul 04 2013: (Start)
%F A020570 a(0)=1, a(1)=21, a(2)=295; for n>2, a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3).
%F A020570 a(n) = 15*a(n-1) - 56*a(n-2) + 6^n. (End)
%F A020570 From _Elmo R. Oliveira_, Mar 26 2025: (Start)
%F A020570 E.g.f.: exp(6*x)*(18 - 49*exp(x) + 32*exp(2*x)).
%F A020570 a(n) = A016170(n+1) - A016169(n+2). (End)
%t A020570 CoefficientList[Series[1/((1-6*x)*(1-7*x)*(1-8*x)), {x, 0, 20}], x]  (* _Harvey P. Dale_, Feb 24 2011 *)
%o A020570 (Magma) m:=20; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-7*x)*(1-8*x)))); // _Vincenzo Librandi_, Jul 04 2013
%o A020570 (Magma) I:=[1, 21, 295]; [n le 3 select I[n] else 21*Self(n-1)-146*Self(n-2)+336*Self(n-3): n in [1..20]]; // _Vincenzo Librandi_, Jul 04 2013
%o A020570 (PARI) my(x='x+O('x^30)); Vec(1/((1-6*x)*(1-7*x)*(1-8*x))) \\ _G. C. Greubel_, Feb 07 2018
%Y A020570 Cf. A016169, A016170.
%K A020570 nonn,easy
%O A020570 0,2
%A A020570 _N. J. A. Sloane_