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A081202 8th binomial transform of (0,1,0,1,0,1,....), A000035.

%I A081202 #25 Aug 27 2025 07:36:09
%S A081202 0,1,16,193,2080,21121,206896,1979713,18640960,173533441,1602154576,
%T A081202 14701866433,134294124640,1222488408961,11099284691056,
%U A081202 100571785292353,909893629141120,8222275592839681,74233110849544336,669726411243809473,6038936596379658400,54430221633714537601
%N A081202 8th binomial transform of (0,1,0,1,0,1,....), A000035.
%C A081202 Binomial transform of A081201.
%C A081202 From _Wolfdieter Lang_, Jul 17 2017: (Start)
%C A081202 For a combinatorial interpretation of a(n) with special 9-letter words of length n see the comment in A081200 on the 7-letter analog.
%C A081202 The binomial transform of {a(n)}_{n >=0} is A081203, the 10-letter analog.
%C A081202 (End)
%H A081202 Vincenzo Librandi, <a href="/A081202/b081202.txt">Table of n, a(n) for n = 0..200</a>
%H A081202 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (16,-63).
%F A081202 a(n) = 16*a(n-1) - 63*a(n-2), a(0)=0, a(1)=1.
%F A081202 G.f.: x/((1-7*x)*(1-9*x)).
%F A081202 a(n) = (9^n - 7^n)/2.
%F A081202 E.g.f.: exp(7*x)*(exp(2*x) - 1)/2. - _Stefano Spezia_, Jul 23 2024
%t A081202 Join[{a=0,b=1},Table[c=16*b-63*a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2011 *)
%t A081202 CoefficientList[Series[x / ((1 - 7 x) (1 - 9 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Aug 07 2013 *)
%o A081202 (Magma) [9^n/2 - 7^n/2: n in [0..25]]; // _Vincenzo Librandi_, Aug 07 2013
%Y A081202 Cf. A000035, A081200, A081201, A081203.
%Y A081202 Apart from offset same as A016178.
%K A081202 easy,nonn,changed
%O A081202 0,3
%A A081202 _Paul Barry_, Mar 11 2003