A081202 - OEIS

%I #23 Jul 23 2024 16:47:11

%S 0,1,16,193,2080,21121,206896,1979713,18640960,173533441,1602154576,

%T 14701866433,134294124640,1222488408961,11099284691056,

%U 100571785292353,909893629141120,8222275592839681,74233110849544336,669726411243809473,6038936596379658400,54430221633714537601

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

%C Binomial transform of A081201.

%C From _Wolfdieter Lang_, Jul 17 2017: (Start)

%C 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 The binomial transform of {a(n)}_{n >=0} is A081203, the 10-letter analog.

%C (End)

%H Vincenzo Librandi, <a href="/A081202/b081202.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)=0, a(1)=1.

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

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

%F E.g.f.: exp(7*x)*(exp(2*x) - 1)/2. - _Stefano Spezia_, Jul 23 2024

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

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

%Y Cf. A000035, A016178, A081200, A081201, A081203.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 11 2003