cp's OEIS Frontend

The value of this continued fraction is the ratio of two Bessel functions: BesselI(0,2)/BesselI(1,2) = A070910/A096789. Or, equivalently, to the ratio of the sums: Sum_{n>=0} 1/(n!n!) and Sum_{n>=0} n/(n!n!). - Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

1.43312...=[1,2,3,4,5,...] = shape of a rectangle which partitions into n squares at stage n; i.e., this is an example of the match between the continued fraction of a number r and a rectangle having shape r. See A188640. - Clark Kimberling, Apr 09 2011

A factorial triangle, with row sums A001040(n+1), n >= 0.

Conjecture: also coefficient triangle of the denominators of the (n-th convergents to) the continued fraction w/(1+w/(2+w/(3+w/... This continued fraction converges to 0.697774657964... = BesselI(1,2)/BesselI(0,2) for w=1. For instance, the denominator of w/(1 + w/(2 + w/(3 + w/(4 + w/5)))) equals 120 + 96*w + 9*w^2. - Wouter Meeussen, Aug 08 2010

For general w, Bill Gosper showed it equals n!*2F3([1/2-n/2,-n/2], [1,-n,-n], 4*w). - Wouter Meeussen, Jan 05 2013

From Wolfdieter Lang, Mar 02 2013: (Start)

The row length sequence of this array is 1 + floor(n/2) = A008619(n), n >= 0.

The continued fraction 0 + K_{k>=1}(x/k) = x/(1+x/(2+x/(3+... has n-th approximation P(n,x)/Q(n,x). These polynomials satisfy the recurrence q(n,x) = n*q(n-1,x) + x*q(n-2,x), for q replaced by P or Q with inputs P(-1,x) = 1, P(0,x) = 0 and Q(-1,x) = 0 and Q(0,1) = 1. The present array provides the Q-coefficients: Q(n,x) = sum(a(n,m)*x^m, m=0 .. floor(n/2)), n >= 0. For the P(n,x)/x coefficients see the companion array A221913. This proves the first part of W. Meeussen's conjecture given above.

The solution with input q(-1,x) = a and q(0,x) = b is then, due to linearity, q(a,b;n,x) = a*P(n,x) + b*Q(n,x). The motivation to look at the q(n,x) recurrence came from an e-mails from Gary Detlefs, who considered integer x and various inputs and gave explicit formulas.

This array coincides with the SW-NE diagonals of the unsigned Laguerre polynomial coefficient triangle |A021009|.

The entries a(n,m) have a combinatorial interpretation in terms of certain so-called labeled Morse code polynomials using dots (length 1) and dashes (of length 2). a(n,m) is the number of possibilities to decorate the n positions 1,2,...,n with m dashes, m from {0, 1, ..., floor(n/2)}, and n-2*m dots. A dot at position k has a weight k and each dash between two neighboring positions has a label x. a(n,m) is the sum of these labeled Morse codes with m dashes after the label x^m has been divided out. E.g., a(5,2) = 5 + 3 + 1 = 9 from the 3 codes: dash dash dot, dash dot dash, and dot dash dash, or (12)(34)5, (12)3(45) and 1(23)(45) with labels (which are in general multiplicative) 5*x^2, 3*x^2 and 1*x^2 , respectively. For the array of these labeled Morse code coefficients see A221915. See the Graham et al. reference, p. 302, on Euler's continuants and Morse code.

Row sums Q(n,1) = A001040(n+1), n >= 0. Alternating row sums Q(n,-1) = A058797(n). (End)

For fixed x the limit of the continued fraction K_{k>=1}(x/k) (see above) can be computed from the large order n behavior of Phat(n,x) and Q(n,x) given in the formula section in terms of Bessel functions. This follows with the well-known large n behavior of BesselI and BesselK, as given, e.g., in the Sidi and Hoggan reference, eqs. (1.1) and (1.2). See also the book by Olver, ch. 10, 7, p. 374. This continued fraction converges for fixed x to sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)). - Wolfdieter Lang, Mar 07 2013

Equivalence to the binomial formula needs formal proof. This c.f. converges to A052119 = 0.697774657964.. = BesselI(1,2)/BesselI(0,2) for w = 1.