A016191 Expansion of g.f. 1/((1-9*x)*(1-12*x)).

1, 21, 333, 4725, 63261, 818181, 10349613, 128978325, 1590786621, 19476859941, 237209103693, 2877890303925, 34817113183581, 420347224031301, 5067043480830573, 61010412902061525, 733977975013590141, 8824412881862748261, 106043049217649978253, 1273867442329472731125

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..900
• Index entries for linear recurrences with constant coefficients, signature (21,-108).

Programs

Mathematica

CoefficientList[Series[1/((1-9x)(1-12x)),{x,0,30}],x] (* or *) LinearRecurrence[{21,-108},{1,21},30] (* Harvey P. Dale, Oct 15 2011 *)

PARI

Vec(1/((1-9*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (12^(n+1) - 9^(n+1))/3. - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005

a(n-1) = Sum_{k=1..n} 3^(n-1)*3^(n-k)*binomial(n,k). - Zerinvary Lajos, Sep 24 2006

From Vincenzo Librandi, Feb 09 2011: (Start)

a(n) = 12*a(n-1) + 9^n, a(0)=1.

a(n) = 21*a(n-1) - 108*a(n-2); a(0)=1, a(1)=21. (End)

E.g.f.: exp(9*x)*(4*exp(3*x) - 3). - Elmo R. Oliveira, Mar 31 2025