A222468 - OEIS

%I #27 Sep 08 2022 08:46:04

%S 1,2,8,36,196,1248,9128,75520,697936,7130400,79830272,972224064,

%T 12798573376,181124475392,2742464277632,44241677392896,

%U 757593444234496,13725165351006720,262293328557596672,5273316901853946880

%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 A222467(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} (2/k))/2 = 1/(1+2/(2+2/(3+2/(4+... is (1/2)*sqrt(2)*BesselI(1,2*sqrt(2))/BesselI(0,2*sqrt(2)) = 0.5631786198117... See A222466 for more decimals.

%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.

%H G. C. Greubel, <a href="/A222468/b222468.txt">Table of n, a(n) for n = 1..445</a>

%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_{m=0..floor(n/2)} a(n-m,m)*2^m, 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) = 2*(w/2)^n*(BesselI(1,w)*BesselK(n+1,w) + BesselK(1,w)*BesselI(n+1,w)*(-1)^(n+1)), n >= 1, where w := -2*sqrt(2).

%F E.g.f.: Pi*(BesselJ(1, 2*i*sqrt(2)*sqrt(1-z))* BesselY(1, 2*i*sqrt(2)) - BesselY(1, (2*i)*sqrt(2)*sqrt(1-z))*BesselJ(1, 2*i*sqrt(2)))/sqrt(1-z) with Bessel functions and the imaginary unit i = sqrt(-1). Phat(0,x) = 0.

%F Asymptotics: lim_{n -> infinity} a(n)/n! = BesselI(1,2*sqrt(2)) /sqrt(2) = 2.3948330992734...

%e a(4) = 4*a(3) + 2*a(2) = 4*8 + 2*2 = 36.

%e Continued fraction convergent: 1/(1+2/(2+2/(3+2/4))) = 9/16 = 36/64 = a(4)/A222467(4).

%e Morse code a(5) = 196 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 = 196.

%t RecurrenceTable[{a[1] == 1, a[2] == 2, a[n] == n*a[n - 1] + 2 a[n - 2]}, a[n], {n, 20}] (* _Ray Chandler_, Jul 30 2015 *)

%o (PARI) m=30; v=concat([1,2], vector(m-2)); for(n=3, m, v[n]=n*v[n-1] +2*v[n-2]); v \\ _G. C. Greubel_, May 17 2018

%o (Magma) I:=[1, 2]; [n le 2 select I[n] else n*Self(n-1) + 2*Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 17 2018

%Y Cf. A084950, A221913, A222467, A001040(n+1) (x=1), A058797 (x=-1).

%K nonn,easy

%O 1,2

%A _Gary Detlefs_ and _Wolfdieter Lang_, Mar 21 2013