A270369 Expansion of g.f. (1-7*x)/(1-9*x).

1, 2, 18, 162, 1458, 13122, 118098, 1062882, 9565938, 86093442, 774840978, 6973568802, 62762119218, 564859072962, 5083731656658, 45753584909922, 411782264189298, 3706040377703682, 33354363399333138, 300189270593998242, 2701703435345984178, 24315330918113857602, 218837978263024718418

Offset: 0

Author

Colin Barker, Mar 18 2016

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (9).

Crossrefs

Cf. A001019 (powers of 9), A054879 (partial sums), A132025.

Cf. similar sequences with g.f. (1-k*x)/(1-9*x) and k=0..8: A001019 (k=0; k=8 gives two initial 1's ), A055275 (k=1), A270472 (k=2), A092810 (k=3), A067403 (k=4), A270473 (k=5), A102518 (k=6), this sequence (k=7).

Programs

Mathematica

CoefficientList[Series[(1-7x)/(1-9x),{x,0,20}],x] (* or *) Join[ {1}, NestList[9#&,2,20]] (* Harvey P. Dale, Oct 15 2017 *)

PARI

Vec((1-7*x)/(1-9*x) + O(x^30))

Formula

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

a(n) = 9*a(n-1) for n>1.

a(n) = 2*9^(n-1) for n>0.

From Amiram Eldar, May 08 2023: (Start)

Sum_{n>=0} 1/a(n) = 25/16.

Sum_{n>=0} (-1)^n/a(n) = 11/20.

Product_{n>=1} (1 - 1/a(n)) = A132025. (End)

E.g.f.: (2*exp(9*x) + 7)/9. - Elmo R. Oliveira, Mar 25 2025