A227524 Expansion of 1/((1-3x)(1-9x)(1-27x)).

1, 39, 1170, 32670, 891891, 24169509, 653373540, 17648258940, 476567558181, 12867905191779, 347438670325110, 9380891170278810, 253284485241566871, 6838684914320250849, 184644527001833063880, 4985402537886183692280, 134605871302457221445961

Offset: 0

Author

Vincenzo Librandi, Jul 17 2013

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (39,-351,729).

Crossrefs

Cf. A016142, A226804.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-9*x)*(1-27*x))));

Magma

I:=[1, 39, 1170]; [n le 3 select I[n] else 39*Self(n-1)-351*Self(n-2)+729*Self(n-3): n in [1..25]];

Mathematica

CoefficientList[Series[1 / ((1 - 3 x) (1 - 9 x) (1 - 27 x)), {x, 0, 30}], x]
LinearRecurrence[{39,-351,729},{1,39,1170},20] (* Harvey P. Dale, Jul 04 2022 *)

Formula

G.f.: 1/((1-3*x)*(1-9*x)*(1-27*x)).

a(n) = 3^n*(3^(n+1)-1)*(3^(n+2)-1)/16.

a(0)=1, a(1)=39, a(2)=1170; for n>2, a(n) = 39*a(n-1)-351*a(n-2)+729*a(n-3).

a(n)-27*a(n-1) = A016142(n), with a(-1)=0; a(n) = A226804(n)-81*A226804(n-1), with A226804(-1)=0. [Bruno Berselli, Jul 17 2013]