A005447 - OEIS

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)

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

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-François 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 */

(SageMath)

@CachedFunction

def h(n): return 1 if (n<1) else ((n+1)/(n+2))*( h(n-1)/n - sum( h(n-j)*h(j)/(j+1) for j in range(1, n) ))

def A005447(n):

if (n<4): return 1

elif (n==4): return -1

else: return numerator(h(n-1))

[A005447(n) for n in range(31)] # G. C. Greubel, Nov 21 2022

CROSSREFS

Cf. A005446 (denominators), A090804/A065973.

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