A222470 Numerator sequence of the n-th convergent of the continued fraction 1/(1-2/(2-2/(3-2/(4-...

1, 2, 4, 12, 52, 288, 1912, 14720, 128656, 1257120, 13571008, 160337856, 2057250112, 28480825856, 423097887616, 6712604550144, 113268081577216, 2025400259289600, 38256068763347968, 761070574748380160

Offset: 1

Author

Gary Detlefs and Wolfdieter Lang, Mar 23 2013

Keywords

• nonn
• easy
• frac

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..oo} (-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.

Examples

			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.
		

Crossrefs

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

Programs

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 *)

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_{m=0..floor(n/2)} a(n-m,m)*(-2)^m, n >= 1, with a(n,m) = binomial(n,m)*(n+1)!/(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: lim_{n->oo} a(n)/n! = BesselJ(1,2*sqrt(2))/(sqrt(2)) = 0.2829799868805...