A052641 Expansion of e.g.f. (1-x)/(1-3*x-x^2+x^3).

1, 2, 14, 132, 1704, 27360, 527760, 11874240, 305343360, 8833224960, 283928198400, 10038995366400, 387222498432000, 16180539927552000, 728132005791590400, 35106736224688128000, 1805508406018437120000

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..375

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 587

Formula

E.g.f.: (1-x)/(1-3*x-x^2+x^3).

a(n+3) = 3*(n+3)*a(n+2) + (6+5*n+n^2)*a(n+1) - (6+11*n+6*n^2+n^3)*a(n).

a(n) = n! * Sum_{alpha=RootOf(1-3*z-z^2+z^3)} (1/74)*(11 + 16*_alpha - 7*alpha^2)*alpha^(-1-n).

a(n) = n!*A030186(n). - R. J. Mathar, Nov 27 2011

Maple

spec:= [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Z), Sequence(Z))))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(coeff(series((1-x)/(1-3*x-x^2+x^3), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Oct 28 2019

Mathematica

With[{nn=20}, CoefficientList[Series[(1-x)/(1-3x-x^2+x^3), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Aug 26 2017 *)

Prog

(PARI) my(x='x+O('x^20)); Vec(serlaplace( (1-x)/(1-3*x-x^2+x^3) )) \\ G. C. Greubel, Oct 28 2019
(Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)/(1-3*x-x^2+x^3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 28 2019
(Sage) [factorial(n)*( (1-x)/(1-3*x-x^2+x^3) ).series(x, n+1).list()[n] for n in (0..20)] # G. C. Greubel, Oct 28 2019
(GAP) a:=[1, 2, 14];; for n in [4..30] do a[n]:=3*(n-1)*a[n-1]+(n-1)*(n-2)*a[n-2]-(n-1)*(6-5*n+n^2)*a[n-3]; od; a; # G. C. Greubel, Oct 28 2019

Crossrefs

Cf. A030186.

Sequence in context: A048990 A089602 A336960 * A157085 A073553 A144097

Adjacent sequences: A052638 A052639 A052640 * A052642 A052643 A052644

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved