A180049 -id:A180049 - 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

A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 8, 0, 9, 0, 1, 0, 33, 0, 14, 0, 1, 48, 0, 87, 0, 20, 0, 1, 0, 279, 0, 185, 0, 27, 0, 1, 384, 0, 975, 0, 345, 0, 35, 0, 1, 0, 2895, 0, 2640, 0, 588, 0, 44, 0, 1, 3840, 0, 12645, 0, 6090, 0, 938, 0, 54, 0, 1, 0, 35685, 0, 41685, 0, 12558, 0, 1422, 0, 65

Offset: 0

Wouter Meeussen, Aug 08 2010

Equivalence to the recurrence formula needs formal proof. This continued fraction converges to 0.525135276160981... for w=1. A conjecture by Ramanujan puts this equal to -1 + 1/(sqrt(e*Pi/2) - Sum_{k>=1} 1/(2k-1)!!).
From Alexander Kreinin, Oct 26 2015: (Start)
Let us denote the continued fraction by U(w).
Then it is easy to show that Mill's ratio, R(w) = (1 - Phi(w))/f(w), where Phi is the standard normal distribution function and f is the standard normal density function, satisfies R(w) = 1/(w + U(w)).
Indeed, R(w) = 1/(w+1/(w+2/(w+3/(w+... Then we find U(w) = 1/R(w) - w. It was proved in Alexander Kreinin (arXiv:1405.5852) that R(w+t) + Q(w, t) = exp(w*t + w^2/2)*R(t), where Q(w,t) = Sum_{k>=0} Sum_{m=0..k} q(k,m) * t^m * w^(k+1)/(k+1)!.
Substituting t=0, we obtain R(w) = exp(w^2/2)*sqrt(Pi/2) - Sum_{n>=0} w^(2n+1)/(2n+1)!!. If w=1 we obtain Ramanujan's formula. (End)

			The denominator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 48 + 87w^2 + 20w^4 + w^6.
From _Joerg Arndt_, Apr 20 2013: (Start)
Triangle begins
     1;
     0,     1;
     2,     0,     1;
     0,     5,     0,     1;
     8,     0,     9,     0,    1;
     0,    33,     0,    14,    0,   1;
    48,     0,    87,     0,   20,   0,   1;
     0,   279,     0,   185,    0,  27,   0,  1;
   384,     0,   975,     0,  345,   0,  35,  0,  1;
     0,  2895,     0,  2640,    0, 588,   0, 44,  0, 1;
  3840,     0, 12645,     0, 6090,   0, 938,  0, 54, 0, 1;
     0, 35685,     0, 41685,    0, ... (End)

G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened
Authors?, Hungarian discussion forum
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 3.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, preprint, 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences 19 (2016), Article 16.6.2.

Cf. A137286, A084950, A180047, A180049.

Cf. A111129.

Table[ CoefficientList[ Denominator[ Together[ Fold[ #2/(w+#1) &, Infinity, Reverse @ Table[ k, {k, 1, n} ] ] ] ], w ], {n, 16} ] (* or equivalently *) Clear[ p ];p[ 0 ]=1; p[ 1 ]=w; p[ n_ ]:=p[ n ]= w*p[ n-1 ] + n*p[ n-2 ]; Table[ CoefficientList[ p[ k ]//Expand, w ], {k,0,15} ]

p(0)=1; p(1)=w; p(n) = w*p(n-1) + n*p(n-2) (conjecture).
T(n,k) = T(n-1,k-1) + n*T(n-2,k), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1. - Philippe Deléham, Oct 28 2013
sum_{k=0..n} T(n,k) = A000932(n). - Philippe Deléham, Oct 28 2013
T(2n,0) = A000165(n); T(2n+1,1) = A129890(n); T(2n+2,2) = A035101(n+2). - Philippe Deléham, Oct 28 2013

A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 6, 1, 0, 24, 6, 0, 120, 36, 1, 0, 720, 240, 12, 0, 5040, 1800, 120, 1, 0, 40320, 15120, 1200, 20, 0, 362880, 141120, 12600, 300, 1, 0, 3628800, 1451520, 141120, 4200, 30, 0, 39916800, 16329600, 1693440, 58800, 630, 1, 0, 479001600

Offset: 0

Wouter Meeussen, Aug 08 2010

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.

			Triangle starts:
  0;
  0,   1;
  0,   2;
  0,   6,   1;
  0,  24,   6;
  0, 120,  36,  1;
  0, 720, 240, 12;
The numerator of w/(1+w/(2+w/(3+w/(4+w/5)))) equals 120*w + 36*w^2 + w^3.

Variant: A221913.

Cf. A084950, A180048, A180049, A008297, A111596, A105278, A052119.

Table[CoefficientList[Numerator[Together[Fold[w/(#2+#1) &,Infinity,Reverse @ Table[k,{k,1,n}]]]],w],{n,16}]; (* or equivalently *) Table[(n-m+1)!/m! *Binomial[n-m,m-1], {n,0,16}, {m,0,Floor[n/2+1/2]}]

T(n,m) = (n-m+1)!/m!*binomial(n-m, m-1) for n >= 0, 0 <= m <= (n+1)/2.

A230698 Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 8, 7, 9, 1, 1, 15, 33, 12, 14, 1, 1, 48, 57, 87, 18, 20, 1, 1, 105, 279, 141, 185, 25, 27, 1, 1, 384, 561, 975, 285, 345, 33, 35, 1, 1, 945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1, 3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1

Offset: 0

Philippe Deléham, Oct 28 2013

Triangle A180048 mixed with triangle A180049.

Let p(n,x) be the polynomial whose coefficients are given by row n; e.g., p(2,x) = 2 + x + x^2; then p(n,x) is the numerator of the rational function given by f(n,x) = x + (n - 1)/f(n-1,x), where f(x,0) = 1. (Sum of numbers in row n) = A000885(n) for n >= 1. (Column 1) = A006882 (n-th term = n!! for n >= 0) - Clark Kimberling, Oct 19 2014

			Triangle begins (0<=k<=n):
1
1, 1
2, 1, 1
3, 5, 1, 1
8, 7, 9, 1, 1
15, 33, 12, 14, 1, 1
48, 57, 87, 18, 20, 1, 1
105, 279, 141, 185, 25, 27, 1, 1
384, 561, 975, 285, 345, 33, 35, 1, 1
945, 2895, 1830, 2640, 510, 588, 42, 44, 1, 1
3840, 6555, 12645, 4680, 6090, 840, 938, 52, 54, 1, 1
10395, 35685, 26685, 41685, 10290, 12558, 1302, 1422, 63, 65, 1, 1

Clark Kimberling, Rows 0..100, flattened

Cf. A180048, A180049.

t[0, 0] = 1; t[1, 0] = 1; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k > n || k < 0, 0, t[n - 1, k - 1] + n*t[n - 2, k]]; Table[t[n, k], {n, 0, 10}, {k, 0, n}](* Clark Kimberling, Oct 19 2014 *)
(* Next, the polynomials *); z = 20; f[x_, n_] := x + n/f[x, n - 1]; f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}]; u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]]  (* A249057 array *)
Flatten[CoefficientList[u, x]] (* A249057 sequence *)
(* Clark Kimberling, Oct 19 2014 *)

T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.
T(n,0) = A006882(n).
T(n+1,1) = A007911(n+3).
Sum_{k=0..n} T(n,k) = A000085(n+1).

Corrected by Clark Kimberling, Oct 21 2014

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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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A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

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A180047 Coefficient triangle of the numerators of the (n-th convergents to) the continued fraction w/(1 + w/(2 + w/(3 + w/(...)))).

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A230698 Triangle read by rows: T(n,k) = T(n-1,k-1) + n*T(n-2,k); T(0,0) = T(1,0) = T(1,1) = 1, T(n,k) = 0 if k>n or if k<0.

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