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%I #64 Jan 01 2024 18:27:36
%S 1,7,42,252,1512,9072,54432,326592,1959552,11757312,70543872,
%T 423263232,2539579392,15237476352,91424858112,548549148672,
%U 3291294892032,19747769352192,118486616113152,710919696678912,4265518180073472,25593109080440832,153558654482644992
%N Expansion of g.f. (1+x)/(1-6*x).
%C Coordination sequence for infinite tree with valency 7.
%C For n >= 1, a(n+1) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3,4,5,6,7} such that for fixed, different x_1, x_2,...,x_n in {1,2,...,n+1} and fixed y_1, y_2,...,y_n in {1,2,3,4,5,6,7} we have f(x_i)<>y_i, (i=1,2,...,n). - _Milan Janjic_, May 10 2007
%C For n >= 1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,2,3,5,6} with no two adjacent letters identical. - _Milan Janjic_, Jan 31 2015
%H Vincenzo Librandi, <a href="/A003949/b003949.txt">Table of n, a(n) for n = 0..1000</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=308">Encyclopedia of Combinatorial Structures 308</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (6).
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F G.f.: (1+x)/(1-6*x).
%F a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 5. - _Philippe Deléham_, Jul 10 2005
%F a(0)=1; for n > 0, a(n) = 7*6^(n-1). - _Vincenzo Librandi_, Nov 18 2010
%F a(0)=1, a(1)=7, a(n) = 6*a(n-1). - _Vincenzo Librandi_, Dec 10 2012
%F E.g.f.: (7*exp(6*x) - 1)/6. - _G. C. Greubel_, Sep 24 2019
%p k:=7; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by _G. C. Greubel_, Sep 24 2019
%t q = 7; Join[{a = 1}, Table[If[n != 0, a = q*a - a, a = q*a], {n, 0, 25}]] (* or *) Join[{1}, 7*6^Range[0, 25]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 11 2011 *)
%t CoefficientList[Series[(1+x)/(1-6*x), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 10 2012 *)
%o (PARI) a(n)=if(n,7*6^(n-1),1) \\ _Charles R Greathouse IV_, Mar 22 2016
%o (Magma) k:=7; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // _G. C. Greubel_, Sep 24 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1+x)/(1-6*x))); // _Marius A. Burtea_, Jan 20 2020
%o (Sage) k=7; [1]+[k*(k-1)^(n-1) for n in (1..25)] # _G. C. Greubel_, Sep 24 2019
%o (GAP) k:=7;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # _G. C. Greubel_, Sep 24 2019
%Y Cf. A003947, A003948, A003950, A003951.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Edited by _N. J. A. Sloane_, Dec 04 2009
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