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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

%I M5399 #61 Nov 22 2022 02:37:06

%S 1,1,1,1,-1,1,1,-139,1,-571,-281,163879,-5221,5246819,5459,-534703531,

%T 91207079,-4483131259,-2650986803,432261921612371,-6171801683,

%U 6232523202521089,4283933145517,-25834629665134204969,11963983648109

%N Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.

%C 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

%C 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.

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

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

%H Vincenzo Librandi, <a href="/A005447/b005447.txt">Table of n, a(n) for n = 0..100</a>

%H J. M. Borwein and R. M. Corless, <a href="http://www.jstor.org/stable/2589743">Emerging Tools for Experimental Mathematics</a>, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.

%H G. Marsaglia and J. C. W. Marsaglia, <a href="http://www.jstor.org/stable/2324749">A new derivation of Stirling's approximation to n!</a>, Amer. Math. Monthly, 97 (1990), 827-829.

%H NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/4.13.E7">Lambert W-Function</a>, section 4.13.7

%H J. C. W. Marsaglia, <a href="http://dx.doi.org/10.1080/00949658608810899">The incomplete gamma function and Ramanujan's rational approximation to exp(x)</a>, J. Statist. Comput. Simulation, 24 (1986), 163-168. [_N. J. A. Sloane_, Jun 23 2011]

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

%e 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 + ...

%p 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:

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

%p seq(A005447(i),i=0..24); # _Peter Luschny_, Feb 08 2011

%p # other program

%p a[1]:=1;

%p M:=25;

%p for n from 2 to M do

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

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

%p a[n]:=t1;

%p od:

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

%t 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 *)

%t 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 *)

%o (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 */

%o (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 */

%o (SageMath)

%o @CachedFunction

%o 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) ))

%o def A005447(n):

%o if (n<4): return 1

%o elif (n==4): return -1

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

%o [A005447(n) for n in range(31)] # _G. C. Greubel_, Nov 21 2022

%Y Cf. A005446 (denominators), A090804/A065973.

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

%K sign,frac

%O 0,8

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Jul 21 2002