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A055505
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Numerators in expansion of (1-x)^(-1/x)/e.
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2
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1, 1, 11, 7, 2447, 959, 238043, 67223, 559440199, 123377159, 29128857391, 5267725147, 9447595434410813, 1447646915836493, 225037938358318573, 29911565062525361, 3651003047854884043877, 38950782815463986767
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Nov 04 2007: (Start) This is also the sequence of numerators associated with expansion of (1+x)^(1/x).
(1 + x)^(1/x)=exp(1)*(1 - 1/2*x + 11/24*x^2 - 7/16*x^3 + 2447/5760*x^4 - 959/2304*x^5 + 238043/580608*x^6 - ...).
(1+x)^(1/x)=exp(log(1+x)/x)=exp(1)*exp(-x/2)*exp(x^2/3)*exp(x^3/4)*...
Let a(n) be this sequence, let b(n) be A055535. Then (1+x)^(1/x)=exp(1)*a(n)/b(n) x^n.
a(n)/b(n) = sum(s(i,i-n)/(i !), i=n,...,infinity),... where s(n,m) is a Stirling number of the first kind.
exp(1)=1+sum(s(i,i)/i !, i=1,... infinity), for the n=1 case.
a(1)/b(1)=1/1 because 1+1/1!+1/2!+1/3!+1/4!+...=exp(1)
a(2)/b(2)=1/2 because 1/2!+3/3!+6/4!+10/5!+...=1/2*exp(1)
a(3)/b(3)=11/24 because 2/3!+11/4!+35/5!+85/6!+...=11/24*exp(1)
a(4)/b(4)=7/16 because 6/4!+50/5!+225/6!+735/7!+...=7/16*exp(1) (End)
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REFERENCES
| Markus Brede, On the convergence of the sequence defining Euler's number, Math. Intelligencer, 27, no. 3 (2005), 6-7.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.
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FORMULA
| See Maple line for formula.
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EXAMPLE
| 1+1/2*x+11/24*x^2+7/16*x^3+2447/5760*x^4+...
1, -1/2, 11/24, -7/16, 2447/5760, -959/2304, 238043/580608, -67223/165888, ...
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MAPLE
| T:=proc(u) local k, l; add( stirling1(u+k, k)*((u+k)!)^(-1)* add( (-1)^l/l!, l=0..u-k), k=0..u); end;
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CROSSREFS
| Cf. A055535, A094638, A130534, A055535.
Sequence in context: A060954 A038321 A002749 * A159526 A090841 A085757
Adjacent sequences: A055502 A055503 A055504 * A055506 A055507 A055508
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KEYWORD
| nonn,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jul 11 2000
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 01 2008 at the suggestion of R. J. Mathar
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