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A005447 Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.

%I A005447 M5399 #61 Nov 22 2022 02:37:06
%S A005447 1,1,1,1,-1,1,1,-139,1,-571,-281,163879,-5221,5246819,5459,-534703531,
%T A005447 91207079,-4483131259,-2650986803,432261921612371,-6171801683,
%U A005447 6232523202521089,4283933145517,-25834629665134204969,11963983648109
%N A005447 Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.
%C A005447 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 A005447 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 A005447 E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.
%D A005447 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005447 Vincenzo Librandi, <a href="/A005447/b005447.txt">Table of n, a(n) for n = 0..100</a>
%H A005447 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 A005447 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 A005447 NIST Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/4.13.E7">Lambert W-Function</a>, section 4.13.7
%H A005447 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 A005447 G.f.: A(x) = Sum_{n>=0} A005447(n)/A005446(n)*x^n satisfies log(A(x)) = A(x) - 1 - x^2/2.
%e A005447 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 A005447 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 A005447 := n -> `if`(n<4,1,`if`(n=4,-1,numer(h(n-1))));
%p A005447 seq(A005447(i),i=0..24); # _Peter Luschny_, Feb 08 2011
%p A005447 # other program
%p A005447 a[1]:=1;
%p A005447 M:=25;
%p A005447 for n from 2 to M do
%p A005447 t1:=a[n-1]/(n+1)-add(a[k]*a[n+1-k],k=2..floor(n/2));
%p A005447 if n mod 2 = 1 then t1:=t1-a[(n+1)/2]^2/2; fi;
%p A005447 a[n]:=t1;
%p A005447 od:
%p A005447 s1:=[seq(a[n],n=1..M)]; # _N. J. A. Sloane_, Jun 23 2011, based on J. Marsaglia's 1986 paper
%t A005447 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 A005447 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 A005447 (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 A005447 (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 A005447 (SageMath)
%o A005447 @CachedFunction
%o A005447 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 A005447 def A005447(n):
%o A005447     if (n<4): return 1
%o A005447     elif (n==4): return -1
%o A005447     else: return numerator(h(n-1))
%o A005447 [A005447(n) for n in range(31)] # _G. C. Greubel_, Nov 21 2022
%Y A005447 Cf. A005446 (denominators), A090804/A065973.
%Y A005447 Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.
%K A005447 sign,frac
%O A005447 0,8
%A A005447 _N. J. A. Sloane_
%E A005447 Edited by _Michael Somos_, Jul 21 2002