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%I #12 Oct 23 2022 01:18:39
%S 1,2,4,12,52,288,1912,14720,128656,1257120,13571008,160337856,
%T 2057250112,28480825856,423097887616,6712604550144,113268081577216,
%U 2025400259289600,38256068763347968,761070574748380160
%N Numerator sequence of the n-th convergent of the continued fraction 1/(1-2/(2-2/(3-2/(4-...
%C The corresponding denominator sequence is A222469(n).
%C 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).
%C 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.
%F Recurrence: a(n) = n*a(n-1) - 2*a(n-2), a(-1) = -1/2, a(0) = 0, n >= 1.
%F 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).
%F 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).
%F 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.
%F Asymptotics: limit(a(n)/n!, n -> infinity) = BesselJ(1,2*sqrt(2))/(sqrt(2)) = 0.2829799868805...
%e a(4) = 4*a(3) - 2*a(2) = 4*4 - 2*2 = 12.
%e Continued fraction convergent: 1/(1-2/(2-2/(3-2/4))) = -3/2 = 12/(-8) = a(4)/A222469(4).
%e 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.
%t 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 *)
%Y Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058798 (x=-1), A222468 (x=2).
%K nonn,easy,frac
%O 1,2
%A _Gary Detlefs_ and _Wolfdieter Lang_, Mar 23 2013
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