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8th binomial transform of (1,0,1,0,1,.....), A059841.
6

%I #21 Sep 08 2022 08:45:09

%S 1,8,65,536,4481,37928,324545,2803256,24405761,213887048,1884629825,

%T 16679193176,148135411841,1319377419368,11777507763905,

%U 105319346802296,943126559710721,8454906106826888,75861524447454785,681125306429182616

%N 8th binomial transform of (1,0,1,0,1,.....), A059841.

%C Binomial transform of A081189.

%C a(n) is also the number of words of length n over an alphabet of nine letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs. for the 1- to 11- letter word cases. For a formulation in terms of maps see a Geoffrey Critzer comment in A081189. - _Wolfdieter Lang_, Jul 17 2017

%H Vincenzo Librandi, <a href="/A081190/b081190.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-63).

%F a(n) = 16*a(n-1) -63*a(n-2), a(0)=1, a(1)=8.

%F G.f.: (1-8*x)/((1-7*x)*(1-9*x)).

%F E.g.f. exp(8*x) * cosh(x).

%F a(n) = 7^n/2 + 9^n/2.

%t CoefficientList[Series[(1 - 8 x) / ((1 - 7 x) (1 - 9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)

%t LinearRecurrence[{16,-63},{1,8},20] (* _Harvey P. Dale_, Apr 04 2017 *)

%o (Magma) [7^n/2 + 9^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013

%Y Cf. A007582, A060531, A081189.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 11 2003