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A084950 Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.

%I A084950 #75 May 23 2025 01:09:21
%S A084950 1,1,2,1,6,4,24,18,1,120,96,9,720,600,72,1,5040,4320,600,16,40320,
%T A084950 35280,5400,200,1,362880,322560,52920,2400,25,3628800,3265920,564480,
%U A084950 29400,450,1,39916800,36288000,6531840,376320,7350,36,479001600,439084800,81648000,5080320,117600,882,1
%N A084950 Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.
%C A084950 A factorial triangle, with row sums A001040(n+1), n >= 0.
%C A084950 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
%C A084950 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
%C A084950 From _Wolfdieter Lang_, Mar 02 2013: (Start)
%C A084950 The row length sequence of this array is 1 + floor(n/2) = A008619(n), n >= 0.
%C A084950 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.
%C A084950 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.
%C A084950 This array coincides with the SW-NE diagonals of the unsigned Laguerre polynomial coefficient triangle |A021009|.
%C A084950 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.
%C A084950 Row sums Q(n,1) = A001040(n+1), n >= 0. Alternating row sums Q(n,-1) = A058797(n). (End)
%C A084950 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
%D A084950 Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; Addison-Wesley, 1994.
%D A084950 F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974 (1991 5th printing).
%H A084950 G. C. Greubel, <a href="/A084950/b084950.txt">Rows, n=0..149 of triangle, flattened</a>
%H A084950 Avram Sidi and Philip E. Hogan, <a href="http://www.ijpam.eu/contents/2011-71-3/16/index.html">Asymptotics of modified Bessel functions of high order</a>. Int. J. of Pure and Appl. Maths. 71 No. 3 (2011) 481-498.
%F A084950 a(n, m) = ((n-m)!/m!)*binomial(n-m,m). - _Wouter Meeussen_, Aug 08 2010
%F A084950 From _Wolfdieter Lang_, Mar 02 2013: (Start)
%F A084950 Recurrence (short version): a(n,m) = n*a(n-1,m) + a(n-2,m-1), n>=1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < 2*m. From the recurrence for the Q(n,x) polynomials given in a comment above.
%F A084950 Recurrence (long version): a(n,m) = (2*(n-m)-1)*a(n-1,m) + a(n-2,m-1) - (n-m-1)^2*a(n-2,m), n >= 1, a(0,0) =1, a(n,-1) = 0,  a(n,m) =0 if n < 2*m. From the standard three term recurrence for the unsigned orthogonal Laguerre polynomials. This recurrence can be simplified to the preceding one, because of the explicit factorial formula given above which follows from the one for the Laguerre coefficients (which, in turn, derives from the Rodrigues formula and the Leibniz rule). This proves the relation a(n,m) = |Lhat(n-m,m)|, with the coefficients |Lhat(n,m)| = |A021009(n,m)| of the unsigned n!*L(n,x) Laguerre polynomials.
%F A084950 For the e.g.f.s of the column sequences see  A021009 (here with different offset, which could be obtained by integration).
%F A084950 E.g.f. for row polynomials gQ(z,x) := Sum_{z>=0} Q(n,x)*z^n = (i*Pi*sqrt(x)/sqrt(1-z))*(BesselJ(1, 2*i*sqrt(x)*sqrt(1-z))*BesselY(0, 2*i*sqrt(x)) - BesselY(1, 2*i*sqrt(x)*sqrt(1-z))*BesselJ(0,2*i*sqrt(x))), with the imaginary unit i = sqrt(-1) and Bessel functions. (End)
%F A084950 The row polynomials are Q(n,x) = Pi*(z/2)^(n+1)*(BesselY(0,z)*BesselJ(n+1,z) - BesselJ(0,z)*BesselY(n+1,z)) with z := -i*2*sqrt(x), and the imaginary unit i. An alternative form is Q(n,x) = 2*(w/2)^(n+1)*(BesselI(0,w)*BesselK(n+1,w) - BesselK(0,w)*BesselI(n+1,w)*(-1)^(n+1)) with w := -2*sqrt(x). See A221913 for the derivation based on Abramowitz-Stegun's Handbook. - _Wolfdieter Lang_, Mar 06 2013
%F A084950 Lim_{n -> infinity} Q(n,x)/n! = BesselI(0,2*sqrt(x)). See a comment on asymptotics above. - _Wolfdieter Lang_, Mar 07 2013
%e A084950 The irregular triangle a(n,m) begins:
%e A084950   n\m          0          1        2        3      4    5  6 ...
%e A084950   O:           1
%e A084950   1:           1
%e A084950   2:           2          1
%e A084950   3:           6          4
%e A084950   4:          24         18        1
%e A084950   5:         120         96        9
%e A084950   6:         720        600       72        1
%e A084950   7:        5040       4320      600       16
%e A084950   8:       40320      35280     5400      200      1
%e A084950   9:      362880     322560    52920     2400     25
%e A084950   10:    3628800    3265920   564480    29400    450    1
%e A084950   11:   39916800   36288000  6531840   376320   7350   36
%e A084950   12:  479001600  439084800 81648000  5080320 117600  882  1
%e A084950 ...Reformatted and extended by _Wolfdieter Lang_, Mar 02 2013
%e A084950 E.g., to get row 7, multiply each term of row 6 by 7, then add the term NW of term in row 6: 5040 = (7)(720); 4320 = (7)(600) + 20; 600 = (7)(72) + 96; 16 = (7)(1) + 9. Thus row 7 = 5040 4320 600 16 with a sum of 9976 = a(7) of A001040.
%e A084950 From _Wolfdieter Lang_, Mar 02 2013: (Start)
%e A084950 Recurrence (short version): a(7,2) = 7*72 + 96 = 600.
%e A084950 Recurrence (long version): a(7,2) = (2*5-1)*72 + 96 - (5-1)^2*9 = 600.
%e A084950 a(7,2) = binomial(5,2)*5!/2! = 10*3*4*5 = 600. (End)
%p A084950 L := (n, k) -> abs(coeff(n!*simplify(LaguerreL(n,x)), x, k)):
%p A084950 seq(seq(L(n-k, k), k=0..n/2), n=0..12); # _Peter Luschny_, Jan 22 2020
%t A084950 Table[CoefficientList[Denominator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k,{k,1,n}]]]],w],{n,16}]; (* _Wouter Meeussen_, Aug 08 2010 *)
%t A084950 (* or equivalently: *)
%t A084950 Table[( (n-m)!*Binomial[n-m, m] )/m! ,{n,0,15}, {m,0,Floor[n/2]}] (* _Wouter Meeussen_, Aug 08 2010 *)
%t A084950 row[n_] := If[n == 0, 1, x/ContinuedFractionK[x, i, {i, 0, n}] // Simplify // Together // Denominator // CoefficientList[#, x] &];
%t A084950 row /@ Range[0, 12] // Flatten (* _Jean-François Alcover_, Oct 28 2019 *)
%Y A084950 Cf. A001040, A180047, A180048, A180049.
%Y A084950 Cf. A021009 (Laguerre triangle). For the A-numbers of the column sequences see the Cf. section of A021009. A221913.
%Y A084950 Cf. A052119.
%K A084950 tabf,nonn,easy
%O A084950 0,3
%A A084950 _Gary W. Adamson_, Jun 14 2003
%E A084950 Rows 12 to 17 added based on formula by _Wouter Meeussen_, Aug 08 2010
%E A084950 Name changed by _Wolfdieter Lang_, Mar 02 2013