A005446 Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.
1, 1, 3, 36, 270, 4320, 17010, 5443200, 204120, 2351462400, 1515591000, 2172751257600, 354648294000, 10168475885568000, 7447614174000, 1830325659402240000, 1595278956070800000, 2987091476144455680000
Offset: 0
Examples
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 + ...
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.
- 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]
Programs
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Maple
Maple program from N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper: 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)];
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Mathematica
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 *)
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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 */
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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 */
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SageMath
@CachedFunction 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) )) def A005446(n): return denominator((-1)^n*b(n)) [A005446(n) for n in range(31)] # G. C. Greubel, Nov 21 2022
Formula
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
Extensions
Edited by Michael Somos, Jul 21 2002
Comments