A020570 Expansion of g.f. 1/((1-6*x)*(1-7*x)*(1-8*x)).

1, 21, 295, 3465, 36751, 365001, 3463615, 31794105, 284628751, 2499039081, 21606842335, 184519243545, 1559982264751, 13079717026761, 108915112739455, 901732722577785, 7429565635164751, 60963378722560041, 498496565225842975, 4064108629664292825, 33049477950757248751

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 (21,-146,336).

Crossrefs

Cf. A016169, A016170.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-6*x)*(1-7*x)*(1-8*x)))); // Vincenzo Librandi, Jul 04 2013

Magma

I:=[1, 21, 295]; [n le 3 select I[n] else 21*Self(n-1)-146*Self(n-2)+336*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 04 2013

Mathematica

CoefficientList[Series[1/((1-6*x)*(1-7*x)*(1-8*x)), {x, 0, 20}], x]  (* Harvey P. Dale, Feb 24 2011 *)

PARI

my(x='x+O('x^30)); Vec(1/((1-6*x)*(1-7*x)*(1-8*x))) \\ G. C. Greubel, Feb 07 2018

Formula

If we define f(m,j,x) = Sum_{k=j..m} (binomial(m,k)*Stirling2(k,j)*x^(m-k)) then a(n-2) = f(n,2,6), (n>=2). - Milan Janjic, Apr 26 2009

a(n) = 18*6^n - 49*7^n + 32*8^n. - R. J. Mathar, Jun 30 2013

From Vincenzo Librandi, Jul 04 2013: (Start)

a(0)=1, a(1)=21, a(2)=295; for n>2, a(n) = 21*a(n-1) - 146*a(n-2) + 336*a(n-3).

a(n) = 15*a(n-1) - 56*a(n-2) + 6^n. (End)

From Elmo R. Oliveira, Mar 26 2025: (Start)

E.g.f.: exp(6*x)*(18 - 49*exp(x) + 32*exp(2*x)).

a(n) = A016170(n+1) - A016169(n+2). (End)