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%I A003951 #59 Sep 08 2022 08:44:32
%S A003951 1,9,72,576,4608,36864,294912,2359296,18874368,150994944,1207959552,
%T A003951 9663676416,77309411328,618475290624,4947802324992,39582418599936,
%U A003951 316659348799488,2533274790395904,20266198323167232,162129586585337856,1297036692682702848
%N A003951 Expansion of g.f.: (1+x)/(1-8*x).
%C A003951 Coordination sequence for infinite tree with valency 9.
%C A003951 Binomial transform is {1, 10, 91, 820, 7381, ...}, see A002452. - _Philippe Deléham_, Jul 22 2005
%C A003951 a(n) equals the number of words of length n on alphabet {0,1,...,8} with no two adjacent letters identical. - _Milan Janjic_, Jan 31 2015 [Corrected by _David Nacin_, May 31 2017]
%H A003951 Vincenzo Librandi, <a href="/A003951/b003951.txt">Table of n, a(n) for n = 0..1000</a>
%H A003951 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=310">Encyclopedia of Combinatorial Structures 310</a>
%H A003951 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A003951 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (8).
%H A003951 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A003951 a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 7. - _Philippe Deléham_, Jul 10 2005
%F A003951 a(0) = 1; for n>0, a(n) = 9*8^(n-1). - _Vincenzo Librandi_, Nov 18 2010
%F A003951 a(0) = 1, a(1) = 9, a(n) = 8*a(n-1). - _Vincenzo Librandi_, Dec 10 2012
%F A003951 E.g.f.: (9*exp(8*x) -1)/8. - _G. C. Greubel_, Sep 24 2019
%p A003951 k:=9; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by _G. C. Greubel_, Sep 24 2019
%t A003951 Join[{1}, 9*8^Range[0, 25]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 11 2011 *)
%t A003951 CoefficientList[Series[(1+x)/(1-8*x), {x, 0, 25}], x] (* _Vincenzo Librandi_, Dec 10 2012 *)
%o A003951 (Magma) [1] cat [9*8^(n-1): n in [1..25]]; // _Vincenzo Librandi_, Dec 11 2012
%o A003951 (PARI) a(n)=if(n,9*8^n/8,1) \\ _Charles R Greathouse IV_, Mar 22 2016
%o A003951 (Sage) k=9; [1]+[k*(k-1)^(n-1) for n in (1..25)] # _G. C. Greubel_, Sep 24 2019
%o A003951 (GAP) k:=9;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # _G. C. Greubel_, Sep 24 2019
%Y A003951 Cf. A003945.
%K A003951 nonn,easy
%O A003951 0,2
%A A003951 _N. J. A. Sloane_
%E A003951 Edited by _N. J. A. Sloane_, Dec 04 2009

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