A053108 - OEIS

%I #28 Oct 01 2023 12:30:31

%S 1,81,3645,120285,3247695,75996063,1595917323,30778405515,

%T 554011299270,9418192087590,152574711818958,2371843247367438,

%U 35577648710511570,517244277406668210,7315311923322878970

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

%H Vincenzo Librandi, <a href="/A053108/b053108.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (81, -2916, 61236, -826686, 7440174, -44641044, 172186884, -387420489, 387420489).

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

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

%F 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

%t CoefficientList[Series[1/(1-9x)^9,{x,0,30}],x] (* _Harvey P. Dale_, Jan 21 2012 *)

%o (Sage)[lucas_number2(n, 9, 0)*binomial(n,8)/9^8 for n in range(8, 23)] # _Zerinvary Lajos_, Mar 13 2009

%o (Magma) [Binomial(n+8, 8)*9^n: n in [0..20]]; // _Vincenzo Librandi_, Oct 13 2011

%o (PARI) vector(20, n, n--; 9^n*binomial(n+8,8)) \\ _G. C. Greubel_, Aug 15 2018

%Y Cf. A053107.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_