0,2

Or, equally, denominators in expansion of (1+x)^(1/x)/e.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

Comments from Miklos Kristof (kristmikl(AT)freemail.hu), Nov 04 2007 (Start): (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 A055505, let b(n) be this sequence. 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)

(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 + ...).

Cf. A094638, A130534, A055505.

Sequence in context: A066585 A075267 A002743 this_sequence A072217 A052686 A064818

Adjacent sequences: A055532 A055533 A055534 this_sequence A055536 A055537 A055538

nonn,frac

N. J. A. Sloane (njas(AT)research.att.com), Jul 11 2000

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 25 2008 at the suggestion of R. J. Mathar and Eric S. Rowland.