A017933 Expansion of 1/((1-3x)(1-6x)(1-9x)).

1, 18, 225, 2430, 24381, 234738, 2205225, 20404710, 186995061, 1703091258, 15448694625, 139763668590, 1262226050541, 11386154248578, 102632111782425, 924629361662070, 8327306431726821, 74979611075290698

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (18,-99,162).

Crossrefs

Third column of triangle A075498.

Cf. A016137, A028085.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-6*x)*(1-9*x)))); /* or */ I:=[1, 18, 225]; [n le 3 select I[n] else 18*Self(n-1)-99*Self(n-2)+162*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 02 2013

Mathematica

CoefficientList[Series[1 / ((1 - 3 x) (1 - 6 x) (1 - 9 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 02 2013 *)
LinearRecurrence[{18,-99,162},{1,18,225},20] (* Harvey P. Dale, Sep 09 2023 *)

PARI

Vec(1/((1-3*x)*(1-6*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (3^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).

a(n) = (3^n - 8*6^n + 9*9^n)/2.

G.f.: 1/((1-3*x)*(1-6*x)*(1-9*x)).

E.g.f.: (d^3/dx^3)((((exp(3*x)-1)/3)^3)/3!) = (exp(3*x) - 8*exp(6*x) + 9*exp(9*x))/2.

a(0)=1, a(1)=18, a(2)=225; for n > 2, a(n) = 18*a(n-1) - 99*a(n-2) + 162*a(n-3). - Vincenzo Librandi, Jul 02 2013

a(n) = 15*a(n-1) - 54*a(n-2) + 3^n. - Vincenzo Librandi, Jul 02 2013