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A222470 Numerator sequence of the n-th convergent of the continued fraction 1/(1-2/(2-2/(3-2/(4-... 4
1, 2, 4, 12, 52, 288, 1912, 14720, 128656, 1257120, 13571008, 160337856, 2057250112, 28480825856, 423097887616, 6712604550144, 113268081577216, 2025400259289600, 38256068763347968, 761070574748380160 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The corresponding denominator sequence is A222469(n).

a(n) = Phat(n,-2) with the numerator polynomials Phat of A221913. All the given formulas follow from there and the comments given under A084950. The limit of the continued fraction (0 + K_{k=1}^infty(2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4-... is (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/ BesselJ(0,2*sqrt(2)) = -1.43974932187023280... (see A222471).

For a combinatorial interpretation in terms of labeled Morse codes see a comment on A221913. Here each dash has label x=-2, and the dots have label j if they are at position j. Labels are multiplied and all codes on [2,...,n+1] are summed.

LINKS

Table of n, a(n) for n=1..20.

FORMULA

Recurrence: a(n) =  n*a(n-1) - 2*a(n-2), a(-1) = -1/2, a(0) = 0, n >= 1.

As a sum:  a(n) = sum(a(n-m,m)*(-2)^m, m =0..floor(n/2)), n >= 1,  with a(n,m) = binomial(n-1-m,m)*(n-m)!/(m+1)!  = |A066667(n,m)| (Laguerre coefficients, parameter alpha = 1).

Explicit form: a(n) = Pi*(z/2)^n*(BesselY(1,z)* BesselJ(n+1,z) - BesselJ(1,z)*BesselY(n+1,z)) with z = 2*sqrt(2).

E.g.f.: Pi*(BesselJ(1, -x*sqrt(1-z))* BesselY(1, -x) - BesselY(1, -x*sqrt(1-z))*BesselJ(1, -x))/sqrt(1-z) with x  = 2*sqrt(x). Here Phat(0,x) = 0.

Asymptotics: limit(a(n)/n!, n -> infinity) = BesselJ(1,2*sqrt(2))/(sqrt(2)) = 0.2829799868805...

EXAMPLE

a(4) = 4*a(3) - 2*a(2) = 4*4 - 2*2 = 12.

Continued fraction convergent: 1/(1-2/(2-2/(3-2/4))) = -3/2 = 12/(-8)  = a(4)/A222469(4).

Morse code a(5) = 52  from the sum of all 5 labeled codes on [2,3,4,5], one with no dash, three with one dash and one with two dashes:  5!/1 + (4*5 + 2*5 + 2*3)*(-2) +(-2)^2 = 52.

MATHEMATICA

Rest[RecurrenceTable[{a[-1]==-(1/2), a[0]==0, a[n]==n*a[n-1]-2a[n-2]}, a, {n, 20}]] (* Harvey P. Dale, Oct 24 2015 *)

CROSSREFS

Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058798 (x=-1), A222468 (x=2).

Sequence in context: A030831 A058767 A075876 * A227037 A158569 A020106

Adjacent sequences:  A222467 A222468 A222469 * A222471 A222472 A222473

KEYWORD

nonn,easy,frac

AUTHOR

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

STATUS

approved

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Last modified September 30 03:35 EDT 2020. Contains 337433 sequences. (Running on oeis4.)