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A005447
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Numerators of expansion of -W_{-1}(-e^{-1-x^2/2}) where W_{-1} is Lambert W function.
(Formerly M5399)
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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REFERENCES
| 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. Simulation, 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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FORMULA
| 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
| 1, 1/3, 1/36, -1/270, 1/4320, 1/17010, -139/5443200, 1/204120, -571/2351462400, ...
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MAPLE
| 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
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
| (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)); 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
| Cf. A005446 (denominators), A090804/A065973.
Sequence in context: A163693 A089518 A199839 * A169599 A047652 A020357
Adjacent sequences: A005444 A005445 A005446 * A005448 A005449 A005450
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KEYWORD
| sign,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Michael Somos, Jul 21, 2002
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