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A005447
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Numerators of the expansion of -W_{-1}(-e^(-1 - x^2/2)) where x > 0 and W_{-1} is the Lambert W function.
(Formerly M5399)
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11
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1, 1, 1, 1, -1, 1, 1, -139, 1, -571, -281, 163879, -5221, 5246819, 5459, -534703531, 91207079, -4483131259, -2650986803, 432261921612371, -6171801683, 6232523202521089, 4283933145517, -25834629665134204969, 11963983648109
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OFFSET
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0,8
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COMMENTS
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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
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.
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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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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.
NIST Digital Library of Mathematical Functions, Lambert W-Function, section 4.13.7
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]
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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, ...
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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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:
A005447 := n -> `if`(n<4, 1, `if`(n=4, -1, numer(h(n-1))));
seq(A005447(i), i=0..24); # Peter Luschny, Feb 08 2011
# other program
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)]; # N. J. A. Sloane, Jun 23 2011, based on J. Marsaglia's 1986 paper
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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 */
(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 */
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CROSSREFS
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Cf. A005446 (denominators), A090804/A065973.
Cf. A299430 / A299431 (A(x)^2), A299432 / A299433.
Sequence in context: A089518 A223192 A199839 * A261703 A266003 A270310
Adjacent sequences: A005444 A005445 A005446 * A005448 A005449 A005450
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KEYWORD
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sign,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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