A081109 9th binomial transform of (1,1,0,0,0,0,0,...).

1, 10, 99, 972, 9477, 91854, 885735, 8503056, 81310473, 774840978, 7360989291, 69735688020, 659002251789, 6213449802582, 58462914051567, 549043018919064, 5147278302366225, 48178524910147866, 450283905890997363

Offset

0,2

Comments

Main diagonal of array defined by m(0,j) = j; m(i,0) = i and m(i,j) = m(i-1,j) + 8*m(i-1,j-1). - Benoit Cloitre, Jun 13 2003

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Index entries for linear recurrences with constant coefficients, signature (18,-81).

Formula

a(n) = 18*a(n-1) - 81*a(n-2), a(0) = 1, a(1) = 10.

a(n) = (n + 9)*9^(n-1).

G.f.: (1 - 8*x)/(1 - 9*x)^2.

E.g.f.: exp(9*x)*(1 + x). - Stefano Spezia, Mar 04 2023

From Amiram Eldar, Jan 04 2026: (Start)

Sum_{n>=0} 1/a(n) = 3486784401 * log(9/8) - 114991534089/280.

Sum_{n>=0} (-1)^n/a(n) = 3486784401 * log(10/9) - 102863432439/280. (End)

Mathematica

CoefficientList[Series[(1 - 8 x) / (1 - 9 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *)

Prog

(Magma) [(n+9)*9^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013

Crossrefs

Cf. A081108, A081122, A006234.

Sequence in context: A224761 A171315 A370043 * A242633 A004189 A322054

Adjacent sequences: A081106 A081107 A081108 * A081110 A081111 A081112

Keyword

easy,nonn

Author

Paul Barry, Mar 07 2003

Status

approved