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

%I #53 Sep 08 2022 08:44:40

%S 1,15,151,1275,9751,70035,481951,3216795,20991751,134667555,852639151,

%T 5343198315,33212784151,205111785075,1260114546751,7708980203835,

%U 46999640806951,285743822630595,1733261544204751

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

%C 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

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

%H Vincenzo Librandi, <a href="/A016103/b016103.txt">Table of n, a(n) for n = 0..200</a>

%H Andi Fugard, <a href="https://www.andifugard.info/wp-content/uploads/2020/08/Fugard-2008-Counting-first-order-models-with-n-individuals-of-syllogisms.pdf">Counting first-order models (with n individuals) of syllogisms</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-74,120).

%F a(n) = 2^(3 + 2*n) + 2^(1 + n) * 3^(2 + n) - 5^(2 + n). - _Andi Fugard_, Jul 22 2008

%F 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

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

%F 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

%F a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3). - _Vincenzo Librandi_, Jun 24 2013

%t CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 6 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Jun 24 2013 *)

%o (PARI) Vec(1/((1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-6*x)))); /* or */ 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

%Y Cf. A051588, A000302, A005060, A003468.

%K nonn,easy

%O 0,2

%A _Robert G. Wilson v_