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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005447 Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.
(Formerly M5399)
11
1, 1, 1, 1, -1, 1, 1, -139, 1, -571, -281, 163879, -5221, 5246819, 5459, -534703531, 91207079, -4483131259, -2650986803, 432261921612371, -6171801683, 6232523202521089, 4283933145517, -25834629665134204969, 11963983648109 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Numerators of the expansion of -W_0(-e^(-1 - x^2/2)) where x < 0 an W_0 is the principal branch of the Lambert W function. - Michael Somos, Oct 06 2017

See A299430/A299431 for more formulas; given g.f. A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n, then A(x)^2 = Sum_{n>=0} A299430(n)/A299431(n)*x^n.

REFERENCES

E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

J. M. Borwein and R. M. Corless, Emerging Tools for Experimental Mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.

G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829.

NIST Digital Library of Mathematical Functions, Lambert W-Function, section 4.13.7

J. C. W. Marsaglia, The incomplete gamma function and Ramanujan's rational approximation to exp(x), J. Statist. Comput. Simulation, 24 (1986), 163-168. [N. J. A. Sloane, Jun 23 2011]

FORMULA

G.f.: A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n satisfies log(A(x)) = A(x) - 1 - x^2/2.

EXAMPLE

1, 1/3, 1/36, -1/270, 1/4320, 1/17010, -139/5443200, 1/204120, -571/2351462400, ...

G.f.: A(x) = 1 + x + 1/3*x^2 + 1/36*x^3 - 1/270*x^4 + 1/4320*x^5 + 1/17010*x^6 - 139/5443200*x^7 + 1/204120*x^8 + ... + A005447(n)/A005446(n)x^n + ...

MAPLE

h := proc(k) option remember; local j; `if`(k<=0, 1, (h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:

A005447 := n -> `if`(n<4, 1, `if`(n=4, -1, numer(h(n-1))));

seq(A005447(i), i=0..24); # Peter Luschny, Feb 08 2011

# other program

a[1]:=1;

M:=25;

for n from 2 to M do

t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k], k=2..floor(n/2));

if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;

a[n]:=t1;

od:

s1:=[seq(a[n], n=1..M)]; # N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper

MATHEMATICA

terms = 25; Assuming[x > 0, -ProductLog[-1, -Exp[-1 - x^2/2]] + O[x]^terms] // CoefficientList[#, x]& // Take[#, terms]& // Numerator (* Jean-Fran├žois Alcover, Jun 20 2013, updated Feb 21 2018 *)

a[ n_] := If[ n < 0, 0, Block[ {$Assumptions = x < 0}, SeriesCoefficient[ -ProductLog[ -Exp[-1 - x^2/2]], {x, 0, n}] // Numerator]]; (* Michael Somos, Oct 06 2017 *)

PROG

(PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n, k, 1); for(k=2, n, A[k] = (A[k-1] - sum(i=2, k-1, i * A[i] * A[k+1-i])) / (k+1)); numerator(A[n]))}; /* Michael Somos, Jun 09 2004 */

(PARI) {a(n) = if( n<1, n==0, numerator( polcoeff( serreverse(sqrt( 2 * (x - log(1 + x + x^2 * O(x^n))))), n)))}; /* Michael Somos, Jun 09 2004 */

CROSSREFS

Cf. A005446 (denominators), A090804/A065973.

Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.

Sequence in context: A089518 A223192 A199839 * A261703 A266003 A270310

Adjacent sequences:  A005444 A005445 A005446 * A005448 A005449 A005450

KEYWORD

sign,frac

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Michael Somos, Jul 21 2002

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 25 20:38 EDT 2019. Contains 321477 sequences. (Running on oeis4.)