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%I M3140 #50 Nov 22 2022 02:32:33
%S 1,1,3,36,270,4320,17010,5443200,204120,2351462400,1515591000,
%T 2172751257600,354648294000,10168475885568000,7447614174000,
%U 1830325659402240000,1595278956070800000,2987091476144455680000
%N Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.
%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="/A005446/b005446.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 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.
%F a(n) = denominator of ((-1)^n * b(n)), where b(n) = (1/(n+1))*( b(n-1) - Sum_{j=2..n-1} j*b(j)*b(n-j+1) ) with b(0) = b(1) = 1 (from Borwein and Corless). - _G. C. Greubel_, Nov 21 2022
%e 1, 1/3, 1/36, -1/270, 1/4320, 1/17010, -139/5443200, 1/204120, -571/2351462400, ...
%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 Maple program from _N. J. A. Sloane_, Jun 23 2011, based on J. Marsaglia's 1986 paper:
%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)];
%t terms = 18; Assuming[x > 0, -ProductLog[-1, -Exp[-1 - x^2/2]] + O[x]^terms] // CoefficientList[#, x]& // Take[#, terms]& // Denominator (* _Jean-François Alcover_, Jun 20 2013, updated Feb 21 2018 *)
%o (PARI) a(n)=local(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)); denominator(A[n])) /* _Michael Somos_, Jun 09 2004 */
%o (PARI) a(n)=if(n<1,n==0,denominator(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 b(n): return 1 if (n<2) else (1/(n+1))*( b(n-1) - sum( j*b(n-j+1)*b(j) for j in range(2,n) ))
%o def A005446(n): return denominator((-1)^n*b(n))
%o [A005446(n) for n in range(31)] # _G. C. Greubel_, Nov 21 2022
%Y Cf. A005447, A090804/A065973.
%Y Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.
%K nonn,frac
%O 0,3
%A _N. J. A. Sloane_
%E Edited by _Michael Somos_, Jul 21 2002
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