Search: id:A055505
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%I A055505
%S A055505 1,1,11,7,2447,959,238043,67223,559440199,123377159,29128857391,
%T A055505 5267725147,9447595434410813,1447646915836493,225037938358318573,
%U A055505 29911565062525361,3651003047854884043877,38950782815463986767
%N A055505 Numerators in expansion of (1-x)^(-1/x)/e.
%C A055505 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).
%C A055505 (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 - ...).
%C A055505 (1+x)^(1/x)=exp(log(1+x)/x)=exp(1)*exp(-x/2)*exp(x^2/3)*exp(x^3/4)*...
%C A055505 Let a(n) be this sequence, let b(n) be A055535. Then (1+x)^(1/x)=exp(1)*a(n)/
               b(n) x^n.
%C A055505 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.
%C A055505 exp(1)=1+sum(s(i,i)/i !, i=1,... infinity), for the n=1 case.
%C A055505 a(1)/b(1)=1/1 because 1+1/1!+1/2!+1/3!+1/4!+...=exp(1)
%C A055505 a(2)/b(2)=1/2 because 1/2!+3/3!+6/4!+10/5!+...=1/2*exp(1)
%C A055505 a(3)/b(3)=11/24 because 2/3!+11/4!+35/5!+85/6!+...=11/24*exp(1)
%C A055505 a(4)/b(4)=7/16 because 6/4!+50/5!+225/6!+735/7!+...=7/16*exp(1) (End)
%D A055505 M. Brede, On the convergence of the sequence defining Euler's number, 
               Math. Intelligencer, 20 (1998), 25-29.
%D A055505 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
%D A055505 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.
%F A055505 See Maple line for formula.
%e A055505 1+1/2*x+11/24*x^2+7/16*x^3+2447/5760*x^4+...
%e A055505 1, -1/2, 11/24, -7/16, 2447/5760, -959/2304, 238043/580608, -67223/165888, 
               ...
%p A055505 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;
%Y A055505 Cf. A055535, A094638, A130534, A055535.
%Y A055505 Sequence in context: A060954 A038321 A002749 this_sequence A159526 A090841 
               A085757
%Y A055505 Adjacent sequences: A055502 A055503 A055504 this_sequence A055506 A055507 
               A055508
%K A055505 nonn,frac
%O A055505 0,3
%A A055505 N. J. A. Sloane (njas(AT)research.att.com), Jul 11 2000
%E A055505 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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