login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Equivalence to the recurrence formula needs formal proof. This C.F. converges to 0.525135276160981.. for w=1. A conjecture by Ramanujan puts this equal to -1+1/(Sqrt[E Pi/2]-Sum[1/(2k-1)!!,{k,1,Infinity}].

Comment 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+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(wt+w^2/2)R(t), where Q(w,t)=Sum[ Sum[q(k,m)t^m w^(k+1)/(k+1)!,{m,0,k}],{k,0,Infinity}].

Substituting t=0, we obtain R(w)=exp(w^2/2) sqrt(Pi/2) - Sum[w^(2n+1)/(2n+1)!!, {n,0,Infinity}]. If w=1 we obtain Ramanujan's formula. (End)

REFERENCES

Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, Preprint 2016; https://www.researchgate.net/profile/Alexander_Kreinin/publication/294260037_Integer_Sequences_and_Laplace_Continued_Fraction/links/56bf5f5308ae2f498ef7effa.pdf

LINKS

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.

FORMULA

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

EXAMPLE

The denominator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 48+87w^2+20w^4+w^6.

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, ...

- Joerg Arndt, Apr 20 2013

MATHEMATICA

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} ]

CROSSREFS

Cf. A137286, A084950, A180047, A180049.

Cf. A111129.

Sequence in context: A238618 A132277 A137286 * A128890 A196777 A078924

Adjacent sequences:  A180045 A180046 A180047 * A180049 A180050 A180051

KEYWORD

nonn,tabl

AUTHOR

Wouter Meeussen, Aug 08 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 23 02:49 EDT 2017. Contains 283901 sequences.