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A084950
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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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16
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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
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OFFSET
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0,3
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COMMENTS
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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. - Wouter Meeussen, Aug 08 2010
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}^{infinity}(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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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.
The denominator of w/(1 + w/(2 + w/(3 + w/(4 + w/5)))) equals 120 + 96w + 9w^2. - Wouter Meeussen, Aug 08 2010
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)
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MAPLE
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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
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MATHEMATICA
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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] &];
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CROSSREFS
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Cf. A021009 (Laguerre triangle). For the A-numbers of the column sequences see the Cf. section of A021009. A221913.
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KEYWORD
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tabf,nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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