A222472 - OEIS

%I #16 Apr 20 2018 11:10:02

%S 1,2,9,42,237,1548,11547,97020,907821,9369270,105785433,1297533006,

%T 17185285377,244486594296,3718854770571,60235136112024,

%U 1035153878216121,18813475216226250,360561490742947113,7267670240507621010

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

%C The corresponding denominator sequence is A213190.

%C a(n) = Phat(n,3) with the numerator polynomials Phat of A221913. All the given formulas follow from there and from the comments given under A084950. The limit of the continued fraction (0 + K_{k=1}^infty(3/k))/3 = 1/(1+3/(2+3/(3+3/(4+... is (1/3)*sqrt(3)*BesselI(1,2*sqrt(3))/BesselI(0,2*sqrt(3)) = 0.484516174987404...

%C For a combinatorial interpretation in terms of labeled Morse codes see a comment on A221913. Here each dash has label x=3, and the dots have label j if they are at position j. Labels are multiplied and all codes on positions [2,...,n+1] are summed.

%F Recurrence: a(n) =  n*a(n-1) + 3*a(n-2), with a(-1) = 1/3, a(0) = 0, n >= 1.

%F As a sum: a(n) = Sum_{m=0..floor(n/2)} b(n-m,m)*3^m, n >= 1,  with b(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*(sqrt(3))^n*(BesselK(1, -2*sqrt(3))* BesselI(n+1, -2*sqrt(3)) + (-1)^(n+1)*BesselI(1,-2*sqrt(3))* BesselK(n+1,-2*sqrt(3))).

%F E.g.f.: Pi*(BesselJ(1, 2*I*sqrt(3)*sqrt(1-z))* BesselY(1, 2*I*sqrt(3)) - BesselY(1, (2*I)*sqrt(3)*sqrt(1-z))*BesselJ(1, 2*I*sqrt(3)))/sqrt(1-z). Here Phat(0,x) = 0.

%F Asymptotics: lim_{n->infinity} a(n)/n! = BesselI(1,2*sqrt(3))/(sqrt(3)) = 3.468649618760...

%e a(4) = 4*a(3) + 2*a(2) = 4*9  + 3*2 = 42.

%e Continued fraction convergent: 1/(1+3/(2+3/(3+3/4))) = 14/29 = 42/87 = a(4)/A213190(4).

%e Morse code: a(5) = 237  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)*(3) +3^2 = 237.

%o (PARI) a=vector(50); a[1]=1;a[2]=2; for(n=3, #a, a[n]=n*a[n-1]+3*a[n-2]); a \\ _Altug Alkan_, Apr 20 2018

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

%K nonn,easy

%O 1,2

%A _Gary Detlefs_ and _Wolfdieter Lang_, Mar 09 2013