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

 

Logo

Invitation: celebrating 50 years of OEIS, 250000 sequences, and Sloane's 75th, there will be a conference at DIMACS, Rutgers, Oct 9-10 2014.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A055535 Denominators in expansion of (1-x)^(-1/x)/e. 4
1, 2, 24, 16, 5760, 2304, 580608, 165888, 1393459200, 309657600, 73574645760, 13377208320, 24103053950976000, 3708162146304000, 578473294823424000, 77129772643123200, 9440684171518279680000, 100969884187361280000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, equally, denominators in expansion of (1+x)^(1/x)/e.

REFERENCES

Chen, Chao-Ping; Choi, Junesang. An Asymptotic Formula for (1+1/x)^x Based on the Partition Function. Amer. Math. Monthly 121 (2014), no. 4, 338--343. MR3183017.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

LINKS

Table of n, a(n) for n=0..17.

FORMULA

From Miklos Kristof, Nov 04 2007 (Start): (1+x)^(1/x) = exp(log(1+x)/x) = exp(1)*exp(-x/2)*exp(x^2/3)*exp(x^3/4)*...

Let a(n) be A055505, let b(n) be this sequence. Then (1+x)^(1/x) = exp(1)*a(n)/b(n) x^n.

a(n)/b(n) = sum(s(i,i-n)/(i !), i = n,...,infinity),... where s(n,m) is a Stirling number of the first kind.

exp(1) = 1+sum(s(i,i)/i !,i = 1,... infinity), for the n = 1 case.

a(1)/b(1) = 1/1 because 1+1/1!+1/2!+1/3!+1/4!+... = exp(1)

a(2)/b(2) = 1/2 because 1/2!+3/3!+6/4!+10/5!+... = 1/2*exp(1)

a(3)/b(3) = 11/24 because 2/3!+11/4!+35/5!+85/6!+... = 11/24*exp(1)

a(4)/b(4) = 7/16 because 6/4!+50/5!+225/6!+735/7!+... = 7/16*exp(1) (End)

EXAMPLE

(1-x)^(-1/x) = exp(1)*(1 + 1/2*x + 11/24*x^2 + 7/16*x^3 + 2447/5760*x^4 + 959/2304*x^5 + 238043/580608*x^6 + ...).

MATHEMATICA

a[n_] := Sum[StirlingS1[n+k, k]/(n+k)!*Sum[(-1)^j/j!, {j, 0, n-k}], {k, 0, n}]; Table[a[n] // Denominator, {n, 0, 17}] (* Jean-Fran├žois Alcover, Mar 04 2014 *)

CROSSREFS

Cf. A094638, A130534, A055505 (numerators).

Sequence in context: A075267 A002743 A220773 * A072217 A229429 A052686

Adjacent sequences:  A055532 A055533 A055534 * A055536 A055537 A055538

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Jul 11 2000

EXTENSIONS

Edited by N. J. A. Sloane, Jul 25 2008 at the suggestion of R. J. Mathar and Eric Rowland

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified September 1 14:28 EDT 2014. Contains 246307 sequences.