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

0, 1, 16, 193, 2080, 21121, 206896, 1979713, 18640960, 173533441, 1602154576, 14701866433, 134294124640, 1222488408961, 11099284691056, 100571785292353, 909893629141120, 8222275592839681, 74233110849544336, 669726411243809473, 6038936596379658400, 54430221633714537601

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• easy
• nonn
• changed

Comments

Binomial transform of A081201.

From Wolfdieter Lang, Jul 17 2017: (Start)

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.

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

(End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (16,-63).

Crossrefs

Cf. A000035, A081200, A081201, A081203.

Apart from offset same as A016178.

Programs

Magma

[9^n/2 - 7^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013

Mathematica

Join[{a=0,b=1},Table[c=16*b-63*a;a=b;b=c,{n,40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011 *)
CoefficientList[Series[x / ((1 - 7 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Aug 07 2013 *)

Formula

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

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

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

E.g.f.: exp(7*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jul 23 2024