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A178690 Expansion of (exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1).

%I A178690 #40 Mar 04 2024 01:14:09
%S A178690 0,0,0,36,432,3660,27000,185556,1223712,7862940,49653000,309776676,
%T A178690 1915868592,11772890220,71992229400,438593697396,2664227115072,
%U A178690 16146540253500,97676540188200,590011376299716,3559691497843152,21455715437760780,129219925869401400
%N A178690 Expansion of (exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1).
%C A178690 a(n) is the number of 3 X n matrices with the following properties:
%C A178690     i) Each row has at least one nonzero entry.
%C A178690    ii) Each column has exactly one nonzero entry.
%C A178690   iii) The nonzero entries in row m, 1 <= m <= 3, are in {1,2,...,m}.
%C A178690 This sequence counts such 3 X n matrices but the results are easily generalized for any such m X n matrix.
%H A178690 Colin Barker, <a href="/A178690/b178690.txt">Table of n, a(n) for n = 0..1000</a>
%H A178690 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (18,-121,372,-508,240).
%F A178690 E.g.f.: (exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1).
%F A178690 G.f.: 12*x^3*(3-18*x+20*x^2)/((1-x)*(1-2*x)*(1-4*x)*(1-5*x)*(1-6*x)). - _Colin Barker_, Nov 30 2014
%F A178690 For n > 0, a(n) = 1 + 2^n - 4^n - 5^n + 6^n. - _Vaclav Kotesovec_, Dec 01 2014
%F A178690 a(n) = 18*a(n-1) - 121*a(n-2) + 372*a(n-3) - 508*a(n-4) + 240*a(n-5). - _Vaclav Kotesovec_, Dec 01 2014
%t A178690 a=Exp[x]-1;b=Exp[2x]-1;c=Exp[3x]-1;Range[0,20]! CoefficientList[Series[a b c,{x,0,20}],x]
%o A178690 (PARI) concat([0,0,0], Vec(-12*x^3*(20*x^2-18*x+3)/((x-1)*(2*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^30))) \\ _Colin Barker_, Dec 01 2014
%o A178690 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (Exp(3*x)-1)*(Exp(2*x)-1)*(Exp(x)-1) )); [0,0,0] cat [Factorial(n+2)*b[n]: n in [1..m-3]]; // _G. C. Greubel_, Jan 26 2019
%o A178690 (Sage) m = 30; T = taylor((exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Jan 26 2019
%Y A178690 Cf. A083321, which is essentially the case for m=2.
%K A178690 nonn,easy
%O A178690 0,4
%A A178690 _Geoffrey Critzer_, Dec 25 2010