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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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11
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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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COMMENTS
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REFERENCES
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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).
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LINKS
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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.
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
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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, ...
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 + ...
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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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MATHEMATICA
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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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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 */
(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))
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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