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A005446
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Denominators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.
(Formerly M3140)
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5
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1, 1, 3, 36, 270, 4320, 17010, 5443200, 204120, 2351462400, 1515591000, 2172751257600, 354648294000, 10168475885568000, 7447614174000, 1830325659402240000, 1595278956070800000, 2987091476144455680000
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OFFSET
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0,3
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REFERENCES
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J. M. Borwein and R. M. Corless, Emerging Tools for Experimental Mathematics, Amer. Math. Monthly, 106 (No. 10, 1999), 889-909.
E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable, 1935, Oxford University Press, p. 221.
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. Similution, 24 (1986), 163-168. [From N. J. A. Sloane, Jun 23 2011]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=0..17.
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FORMULA
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G.f.: A(x)=Sum_{n>=0} A005447(n)/A005446(n)x^n satisfies log(A(x))=A(x)-1-x^2/2.
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EXAMPLE
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1, 1/3, 1/36, -1/270, 1/4320, 1/17010, -139/5443200, 1/204120, -571/2351462400, ...
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MAPLE
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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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PROG
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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 */
(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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CROSSREFS
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Cf. A005447, A090804/A065973.
Sequence in context: A188891 A073992 A127960 * A056307 A056299 A212616
Adjacent sequences: A005443 A005444 A005445 * A005447 A005448 A005449
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KEYWORD
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nonn,frac
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Edited by Michael Somos, Jul 21, 2002
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STATUS
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approved
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