A019677 Expansion of 1/((1-4x)(1-8x)(1-12x)).

1, 24, 400, 5760, 77056, 989184, 12390400, 152862720, 1867841536, 22682271744, 274333696000, 3309180026880, 39847582498816, 479270434504704, 5760041038643200, 69190860134154240, 830853267268304896, 9974742789667160064, 119732942204305408000

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 (24,-176,384).

Crossrefs

Third column of triangle A075499.

Cf. A016152, A075907.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-8*x)*(1-12*x)))); /* or */ I:=[1, 24, 400]; [n le 3 select I[n] else 24*Self(n-1)-176*Self(n-2)+384*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 03 2013

Maple

a:= n-> (Matrix(3, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [24, -176, 384][i], 0)))^n)[1, 1]: seq(a(n), n=0..25);  # Alois P. Heinz, Jul 03 2013

Mathematica

CoefficientList[Series[1 / ((1 - 4 x) (1 - 8 x) (1 - 12 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 03 2013 *)
LinearRecurrence[{24,-176,384},{1,24,400},20] (* Harvey P. Dale, Jul 18 2020 *)

PARI

Vec(1/((1-4*x)*(1-8*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (4^n)*Stirling2(n+3, 3), n >= 0, with Stirling2(n, m) = A008277(n, m).

a(n) = (4^n - 8*8^n + 9*12^n)/2.

G.f.: 1/((1-4*x)*(1-8*x)*(1-12*x)).

E.g.f.: (d^3/dx^3)((((exp(4*x)-1)/4)^3)/3!) = (exp(4*x) - 8*exp(8*x) + 9*exp(12*x))/2.

a(0)=1, a(1)=24, a(2)=400; for n > 2, a(n) = 24*a(n-1) - 176*a(n-2) + 384*a(n-3). - Vincenzo Librandi, Jul 03 2013

a(n) = 30*a(n-1) - 96*a(n-2) + 4^n. - Vincenzo Librandi, Jul 03 2013