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Expansion of 1/((1-6*x)*(1-10*x)*(1-11*x)).
0

%I #34 Mar 10 2020 10:28:02

%S 1,27,493,7599,106645,1411431,17955757,222093423,2690508229,

%T 32080473975,377794514461,4405195463487,50953884924853,

%U 585473143132359,6690087028209805,76090252032830991,861988540696279717

%N Expansion of 1/((1-6*x)*(1-10*x)*(1-11*x)).

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (27,-236,660).

%F a(n) = 21*a(n-1) - 110*a(n-2) + 6^n for n>1, a(0)=1, a(1)=27. - _Vincenzo Librandi_, Mar 11 2011

%F a(n) = (4*11^(n+2) - 5*10^(n+2) + 6^(n+2))/20. - _Yahia Kahloune_, Jun 30 2013

%F In general, for the expansion of 1/((1-r*x)(1-s*x)(1-t*x)) with t > s > r, we have the formula: a(n) = ((s-r)*t^(n+2) - (t-r)*s^(n+2) + (t-s)*r^(n+2))/((s-r)*(t-r)*(t-s)). - _Yahia Kahloune_, Sep 09 2013

%F a(0) = 1, a(1) = 27, a(2) = 493, a(n) = 27*a(n-1) - 236*a(n-2) + 660*a(n-3). - _Harvey P. Dale_, Oct 01 2014

%t CoefficientList[Series[1/((1-6x)(1-10x)(1-11x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-236,660},{1,27,493},30] (* _Harvey P. Dale_, Oct 01 2014 *)

%o (PARI) Vec(1/((1-6*x)*(1-10*x)*(1-11*x)) + O(x^30)) \\ _Jinyuan Wang_, Mar 10 2020

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_