A221915 -id:A221915 - cp's OEIS frontend

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.

1, 1, 2, 1, 6, 4, 24, 18, 1, 120, 96, 9, 720, 600, 72, 1, 5040, 4320, 600, 16, 40320, 35280, 5400, 200, 1, 362880, 322560, 52920, 2400, 25, 3628800, 3265920, 564480, 29400, 450, 1, 39916800, 36288000, 6531840, 376320, 7350, 36, 479001600, 439084800, 81648000, 5080320, 117600, 882, 1

Offset: 0

Gary W. Adamson, Jun 14 2003

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

			The irregular triangle a(n,m) begins:
  n\m          0          1        2        3      4    5  6 ...
  O:           1
  1:           1
  2:           2          1
  3:           6          4
  4:          24         18        1
  5:         120         96        9
  6:         720        600       72        1
  7:        5040       4320      600       16
  8:       40320      35280     5400      200      1
  9:      362880     322560    52920     2400     25
  10:    3628800    3265920   564480    29400    450    1
  11:   39916800   36288000  6531840   376320   7350   36
  12:  479001600  439084800 81648000  5080320 117600  882  1
...Reformatted and extended by _Wolfdieter Lang_, Mar 02 2013
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.
From _Wolfdieter Lang_, Mar 02 2013: (Start)
Recurrence (short version): a(7,2) = 7*72 + 96 = 600.
Recurrence (long version): a(7,2) = (2*5-1)*72 + 96 - (5-1)^2*9 = 600.
a(7,2) = binomial(5,2)*5!/2! = 10*3*4*5 = 600. (End)

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; Addison-Wesley, 1994.
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974 (1991 5th printing).

G. C. Greubel, Rows, n=0..149 of triangle, flattened
Avram Sidi and Philip E. Hogan, Asymptotics of modified Bessel functions of high order. Int. J. of Pure and Appl. Maths. 71 No. 3 (2011) 481-498.

Cf. A001040, A180047, A180048, A180049.
Cf. A021009 (Laguerre triangle). For the A-numbers of the column sequences see the Cf. section of A021009. A221913.
Cf. A052119.

L := (n, k) -> abs(coeff(n!*simplify(LaguerreL(n,x)), x, k)):
seq(seq(L(n-k, k), k=0..n/2), n=0..12); # Peter Luschny, Jan 22 2020

Table[CoefficientList[Denominator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k,{k,1,n}]]]],w],{n,16}]; (* Wouter Meeussen, Aug 08 2010 *)
(* or equivalently: *)
Table[( (n-m)!*Binomial[n-m, m] )/m! ,{n,0,15}, {m,0,Floor[n/2]}] (* Wouter Meeussen, Aug 08 2010 *)
row[n_] := If[n == 0, 1, x/ContinuedFractionK[x, i, {i, 0, n}] // Simplify // Together // Denominator // CoefficientList[#, x] &];
row /@ Range[0, 12] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

a(n, m) = ((n-m)!/m!)*binomial(n-m,m). - Wouter Meeussen, Aug 08 2010
From Wolfdieter Lang, Mar 02 2013: (Start)
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.
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.
For the e.g.f.s of the column sequences see  A021009 (here with different offset, which could be obtained by integration).
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)
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
Lim_{n -> infinity} Q(n,x)/n! = BesselI(0,2*sqrt(x)). See a comment on asymptotics above. - Wolfdieter Lang, Mar 07 2013

Rows 12 to 17 added based on formula by Wouter Meeussen, Aug 08 2010

Name changed by Wolfdieter Lang, Mar 02 2013

A221913 Array of coefficients of numerator polynomials (divided by x) of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+...

Original entry on oeis.org

1, 2, 6, 1, 24, 6, 120, 36, 1, 720, 240, 12, 5040, 1800, 120, 1, 40320, 15120, 1200, 20, 362880, 141120, 12600, 300, 1, 3628800, 1451520, 141120, 4200, 30, 39916800, 16329600, 1693440, 58800, 630, 1, 479001600, 199584000, 21772800, 846720, 11760, 42

Offset: 1

Wolfdieter Lang, Feb 23 2013

The row length sequence of this array is 1 + floor((n-1)/2) = A008619(n-1), n >= 1.
The array of denominators is found under A084950.
The continued fraction 0 + K_{k=1..infinity}(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 coefficients for Phat(n,x) := P(n,x)/x = sum(a(n,m)*x^m,m=0..floor((n-1)/2)), n >= 1. The recurrence is that of q(n,x) and the inputs are Phat(-1,x) = 1/x and Phat(0,x) =0. For the Q(n,x) coefficients see the companion array 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*x*Phat(n,x) + b*Q(n,x).  The motivation to consider the q(n,x) recurrence stems from 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 coefficient array |A066667| or A105278 (taken with offset [0,0]) of the generalized Laguerre polynomials n!*L(1,n,x) (parameter alpha = 1).
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-1 positions 2,...,n with m dashes, m from {0,1,...,floor((n-1)/2)}, and n-1-2*m dots. A dot at position k has a label 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(6,2) = 6 + 4 + 2 = 12  from the 3 codes:  dash dash dot, dash dot dash,and  dot dash dash, or (23)(45)6, (23)4(56) and 2(34)(56), and labels (which are in general multiplicative)  6*x^2, 4*x^2 and 2*x^2, respectively.
For general Morse code polynomials (Euler's continuants) see the Graham et al. reference given in A221915, p. 302. - Wolfdieter Lang, Feb 28 2013
Row sums Phat(n,1) = A001053(n+1), n >= 1. Alternating row sums  Phat(n,-1) = A058798(n), n >= 1.
From Wolfdieter Lang, Mar 06 2013 (Start)
The recurrence for q(n,x) given above, can be transformed to the one of Bessel functions given in Abramowitz-Stegun (see A103921 for the reference) in the first line of eq. 9.1.27 on p. 361 via i^n*q(n,x)/sqrt(x)^n = C(n+1,-i*2*sqrt(x)) with the imaginary unit i, where C can stand for BesselJ or BesselY. In order to fix the two inputs for the Q or Phat polynomials (given above) one uses a linear combination of these two independent solutions. The Wronskian eq. 9.1.16, p. 360, is used to simplify the coefficients. One can also use an alternative version based on eqs. 9.6.3 and 9.6.5, p. 375, to trade the J and Y polynomials for I and K.
  This produces the two explicit formulas given below, and also the two versions given for Q in A084950.
(End)
For large order n the behavior of the row polynomials Phat(n,x) (see above) is known from the one of Bessel functions. See a comment on asymptotics under A084950. This leads then to the limit for Phat(n,x)/n! given in the formula section. The limit for the continued fraction mentioned in the name and above is also found in this comment on A084950. - Wolfdieter Lang, Mar 08 2013
This is the unsigned Lah triangle read by ascending antidiagonals. Conversely, reading the given triangle beginning at the left in descending steps yields a row of the unsigned Lah triangle. This can be verified immediately by means of the explicit formulas. For example, [T(5,0), T(6,1), T(7,2), T(8,3), T(9,4)] is row 5 of A105278. - Peter Luschny, Dec 07 2019

			The irregular triangle a(n,m)  begins:
n\m          0          1         2        3      4    5  6
1:           1
2:           2
3:           6          1
4:          24          6
5:         120         36         1
6:         720        240        12
7:        5040       1800       120        1
8:       40320      15120      1200       20
9:      362880     141120     12600      300     1
10:    3628800    1451520    141120     4200    30
11:   39916800   16329600   1693440    58800    63     1
12:  479001600   19958400  21772800   846720  11760   42
13: 6227020800 2634508800 299376000 12700800 211680 1176  1
...
Recurrence (short version): a(6,1) = 6*36 + 24 = 240.
Recurrence (long version): a(6,1) = 2*4*36 + 24 - 4*3*6 = 240.
a(6,1) = binomial(4,1)*5!/2! = 4*3*4*5 = 240.

Cf. |A066667|, A105278, A084950, A221915.

row[n_] := x/ContinuedFractionK[x, i, {i, 0, n}] // Simplify // Together // Numerator // CoefficientList[#, x]& // Rest;
row /@ Range[12] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

Recurrence (short version): a(n,m) = n*a(n-1,m)  + a(n-2,m-1), n>=2, a(1,1) =1, a(n,-1) = 0, a(n,m) = 0 if n < 2*m+1. From the recurrence for the Phat(n,x) polynomials given in a comment above.
Recurrence (long version): a(n,m) = 2*(n-1-m)*a(n-1,m) + a(n-2,m-1) - (n-1-m)*(n-2-m)*a(n-2,m), n >= 1, a(1,0) = 1, a(n,-1) = 0, a(n,m) = 0 if  n < 2*m + 1. From the recurrence for the unsigned  generalized Laguerre polynomial with parameter alpha = 1. This recurrence can be simplified to the preceding short version, because the following explicit form follows from the one for the generalized Laguerre coefficients (which, in turn, derives from the Rodrigues formula and the Leibniz rule). This proves the relation a(n,m) = |Lhat(1,n-1-m,m)|, with the coefficients |Lhat(1,n,m)| = |A066667(n,m)|  of the unsigned n!*L(1,n,x) Laguerre polynomials (parameter alpha = 1).
a(n,m) = binomial(n-1-m,m)*(n-m)!/(m+1)!, n >= 1, 0 <= m <= floor((n-1)/2).
For the e.g.f.s of the column sequences see A105278 (here with different offset, which could be obtained by integration).
E.g.f. for row polynomials  gPhat(z,x) := Sum_{z>=0} Phat(n,x)*z^n = Pi*(BesselJ(1, 2*i*sqrt(x)*sqrt(1-z))*BesselY(1, 2*i*sqrt(x)) - BesselY(1, (2*i)*sqrt(x)*sqrt(1-z))*BesselJ(1, 2*i*sqrt(x)))/sqrt(1-z) with Bessel functions and the imaginary unit i = sqrt(-1). Phat(0,x) = 0.
From Wolfdieter Lang, Mar 06 2013 (Start)
For the row polynomials one finds Phat(n,x) = Pi*(z/2)^n*(BesselY(1,z)* BesselJ(n+1,z) - BesselJ(1,z)*BesselY(n+1,z)) where z := -i*2*sqrt(x) and the i is the imaginary unit. An alternative form is Phat(n,x) = 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(x). See a comment above for the derivation. (End)
Limit_{n -> oo} Phat(n,x)/n! = BesselI(1,2*sqrt(x))/sqrt(x). See a comment above. - Wolfdieter Lang, Mar 08 2013

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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.

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