A053108 Expansion of 1/(1 - 9*x)^9.

1, 81, 3645, 120285, 3247695, 75996063, 1595917323, 30778405515, 554011299270, 9418192087590, 152574711818958, 2371843247367438, 35577648710511570, 517244277406668210, 7315311923322878970

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (81, -2916, 61236, -826686, 7440174, -44641044, 172186884, -387420489, 387420489).

Crossrefs

Cf. A053107.

Programs

Magma

[Binomial(n+8, 8)*9^n: n in [0..20]]; // Vincenzo Librandi, Oct 13 2011

Mathematica

CoefficientList[Series[1/(1-9x)^9,{x,0,30}],x] (* Harvey P. Dale, Jan 21 2012 *)

PARI

vector(20, n, n--; 9^n*binomial(n+8,8)) \\ G. C. Greubel, Aug 15 2018

Sage

[lucas_number2(n, 9, 0)*binomial(n,8)/9^8 for n in range(8, 23)] # Zerinvary Lajos, Mar 13 2009

Formula

a(n) = 9^n*binomial(n+8, 8).

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

a(n) = 81*a(n-1) - 2916*a(n-2) + 61236*a(n-3) - 826686*a(n-4) + 7440174*a(n-5) - 44641044*a(n-6) + 172186884*a(n-7) - 387420489*a(n-8) + 387420489*a(n-9); a(0)=1, a(1)=81, a(2)=3645, a(3)=120285, a(4)=3247695, a(5)=75996063, a(6)=1595917323, a(7)=30778405515, a(8)=554011299270. - Harvey P. Dale, Jan 21 2012