A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

1, 15, 151, 1275, 9751, 70035, 481951, 3216795, 20991751, 134667555, 852639151, 5343198315, 33212784151, 205111785075, 1260114546751, 7708980203835, 46999640806951, 285743822630595, 1733261544204751

Offset: 0

Author

Robert G. Wilson v

Keywords

• nonn
• easy

Comments

2*a(n-2) = 6^n - 2*5^n + 4^n is the number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) first and third row have a common 1, (c) second and third row have no common 1. - Andi Fugard and Vladeta Jovovic, Jul 26 2008

This is the third column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See A193685 for general comments. - Wolfdieter Lang, Oct 08 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Andi Fugard, Counting first-order models (with n individuals) of syllogisms.
• Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

Crossrefs

Cf. A051588, A000302, A005060, A003468.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-6*x)))); // Vincenzo Librandi, Jun 24 2013

Magma

I:=[1, 15, 151]; [n le 3 select I[n] else 15*Self(n-1)-74*Self(n-2)+120*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 24 2013

Mathematica

CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 6 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

PARI

Vec(1/((1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = 2^(3 + 2*n) + 2^(1 + n) * 3^(2 + n) - 5^(2 + n). - Andi Fugard, Jul 22 2008

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,2,4), n >= 2. - Milan Janjic, Apr 26 2009

O.g.f.: 1/((1-4*x)*(1-5*x)*(1-6*x)).

E.g.f.: (d^2/dx^2)(exp(4*x)*((exp(x)-1)^2)/2!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 08 2011

a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3). - Vincenzo Librandi, Jun 24 2013