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A052124 E.g.f.: exp(-2x)/(1-x)^3. 4
1, 1, 4, 16, 88, 568, 4288, 36832, 354688, 3781504, 44199424, 561823744, 7714272256, 113769309184, 1793341407232, 30085661765632, 535170830467072, 10060645294440448, 199287423535808512, 4148644277780217856 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64(b).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

a(n) = n*a(n-1) + 2*(n-1)*a(n-2). - Detlef Pauly (dettodet(AT)yahoo.de), Sep 22 2003

a(n) = (n+5)*(n+2)*n! * Sum_{k=0..n} (-1)^k*2^(k+2)*(k+3)/(k+5)!. - Vaclav Kotesovec, Oct 28 2012

G.f.: 1/Q(0), where Q(k)=  1 + 2*x - x*(k+3)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

a(n) ~ n!*(n+5)*(n+2)/(2*exp(2)). - Vaclav Kotesovec, Jun 15 2013

From Peter Bala, Sep 20 2013: (Start)

a(n) ~ 1/2*n^2*n!*1/e^2 for large n.

Letting n -> infinity in the above series for a(n) given by Kotesovec gives the series expansion 1/e^2 = Sum_{k >= 0} (-1)^k*(k + 3)*2^(k + 3)/(k + 5)!.

The sequence b(n) := 1/2*n!*(n + 2)*(n + 5) satisfies the recurrence for a(n) given above by Pauly but with the starting values b(0) = 5 and b(1) = 9. This leads to the finite continued fraction expansion a(n) = 1/2*n!*(n + 2)*(n + 5)( 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/n)))) ), valid for n >= 2. Letting n -> infinity in the previous result gives the infinite continued fraction expansion 1/e^2 = 1/(5 + 4/(1 + 2/(2 + 4/(3 + ... + 2*(n-1)/(n + ...))))). Cf. A082031. (End)

MAPLE

A052124 := proc(n) option remember; if n <=1 then 1 else n*A052124(n-1)+2*(n-1)*A052124(n-2); fi; end; # Detlef Pauly

MATHEMATICA

Table[(n+5)*(n+2)*n!*Sum[(-1)^k*2^(k+2)*(k+3)/(k+5)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 28 2012 *)

With[{nn=20}, CoefficientList[Series[Exp[(-2x)]/(1-x)^3, {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Oct 23 2017 *)

CROSSREFS

Cf. A052127, A082031.

Sequence in context: A165964 A005618 A005495 * A235166 A013030 A124962

Adjacent sequences:  A052121 A052122 A052123 * A052125 A052126 A052127

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 23 2000

STATUS

approved

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Last modified December 15 03:08 EST 2017. Contains 296020 sequences.