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%I #39 Dec 27 2024 02:52:02
%S 1,22,313,3666,38493,377286,3529681,31947322,282198565,2447183310,
%T 20920905369,176852694018,1481626607917,12322682753494,
%U 101879323774177,838170485025354,6867569457133749,56077266261254238
%N Expansion of 1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)).
%C From _Bruno Berselli_, May 09 2013: (Start)
%C a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);
%C a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);
%C a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);
%C a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);
%C a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);
%C a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);
%C a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);
%C a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);
%C a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);
%C a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);
%C a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);
%C a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);
%C a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);
%C a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),
%C and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see _Harvey P. Dale_ in Formula lines). (End)
%H Vincenzo Librandi, <a href="/A025992/b025992.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 (22,-171,542,-560).
%F a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - _Harvey P. Dale_, Jan 29 2013
%F a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - _Yahia Kahloune_, May 07 2013
%F E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - _G. C. Greubel_, Dec 27 2024
%t CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* _Harvey P. Dale_, Jan 29 2013 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // _Bruno Berselli_, May 09 2013
%o (PARI) a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ _Charles R Greathouse IV_, Oct 03 2016
%o (Python)
%o def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
%o print([A025992(n) for n in range(41)]) # _G. C. Greubel_, Dec 27 2024
%Y Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_