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

0, 0, 0, 36, 432, 3660, 27000, 185556, 1223712, 7862940, 49653000, 309776676, 1915868592, 11772890220, 71992229400, 438593697396, 2664227115072, 16146540253500, 97676540188200, 590011376299716, 3559691497843152, 21455715437760780, 129219925869401400

Offset: 0

Author

Geoffrey Critzer, Dec 25 2010

Keywords

• nonn
• easy

Comments

a(n) is the number of 3 X n matrices with the following properties:

i) Each row has at least one nonzero entry.

ii) Each column has exactly one nonzero entry.

iii) The nonzero entries in row m, 1 <= m <= 3, are in {1,2,...,m}.

This sequence counts such 3 X n matrices but the results are easily generalized for any such m X n matrix.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (18,-121,372,-508,240).

Crossrefs

Cf. A083321, which is essentially the case for m=2.

Programs

Magma

m:=30; R:=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

Mathematica

a=Exp[x]-1;b=Exp[2x]-1;c=Exp[3x]-1;Range[0,20]! CoefficientList[Series[a b c,{x,0,20}],x]

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

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

Formula

E.g.f.: (exp(3*x)-1)*(exp(2*x)-1)*(exp(x)-1).

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

For n > 0, a(n) = 1 + 2^n - 4^n - 5^n + 6^n. - Vaclav Kotesovec, Dec 01 2014

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