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A052669 Expansion of e.g.f. (1-2*x)/(1-3*x-x^2+2*x^3). 1
1, 1, 8, 66, 840, 12960, 242640, 5286960, 131765760, 3693755520, 115058361600, 3942342835200, 147360531225600, 5967185903078400, 260221271108198400, 12158477739023616000, 605960547270414336000, 32087688283562655744000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..350

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 617

FORMULA

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

Recurrence: a(0)=1, a(1)=1, a(2)=8, a(n) = 3*n*a(n-1) + n*(n-1)*a(n-2) - 2*n*(n-1)*(n-2)*a(n-3).

a(n) = (n!/229)*Sum_{alpha=RootOf(1 - 3*Z - Z^2 + 2*Z^3)} (5 + 74*alpha - 16*alpha^2)*alpha^(-1-n).

a(n) = n!*A052550(n). - R. J. Mathar, Nov 27 2011

MAPLE

spec := [S, {S=Sequence(Prod(Z, Union(Z, Sequence(Union(Z, Z)))))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);

MATHEMATICA

b[n_]:= b[n]= If[n<3, 1+3*Floor[n/2], 3*b[n-1] +b[n-2] -2*b[n-3]];

A052669[n_] := n!*b[n]; (* b = A052550 *)

Table[A052669[n], {n, 0, 40}] (* G. C. Greubel, Jun 14 2022 *)

PROG

(Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( (1-2*x)/(1-3*x-x^2+2*x^3) ))); // G. C. Greubel, Jun 14 2022

(SageMath)

@CachedFunction

def b(n): # b = A052550

    if (n<3): return 1 + 3*(n//2)

    else: return 3*b(n-1) +b(n-2) -2*b(n-3)

def A052669(n): return factorial(n)*b(n)

[A052669(n) for n in (0..40)] # G. C. Greubel, Jun 14 2022

CROSSREFS

Cf. A000142, A052550.

Sequence in context: A188450 A121766 A052620 * A076527 A247112 A166815

Adjacent sequences:  A052666 A052667 A052668 * A052670 A052671 A052672

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)