A055505 Numerators in expansion of (1-x)^(-1/x)/e.

1, 1, 11, 7, 2447, 959, 238043, 67223, 559440199, 123377159, 29128857391, 5267725147, 9447595434410813, 1447646915836493, 225037938358318573, 29911565062525361, 3651003047854884043877, 38950782815463986767

Offset: 0

Author

N. J. A. Sloane, Jul 11 2000

Keywords

• nonn
• frac

Comments

From Miklos Kristof, 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_{i>=n} s(i,i-n)/i! where s(n,m) is a Stirling number of the first kind.

exp(1) = 1 + Sum_{i>=1} s(i,i)/i!, 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)

Examples

			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, ...
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
• Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• Markus Brede, On the convergence of the sequence defining Euler's number, Math. Intelligencer, 27, no. 3 (2005), 6-7.
• Chao-Ping Chen and Junesang Choi, An Asymptotic Formula for (1+1/x)^x Based on the Partition Function, Amer. Math. Monthly 121 (2014), no. 4, 338--343. MR3183017.
• Branko Malesevic, Yue Hu, and Cristinel Mortici, Accurate Estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit, arXiv:1801.04963 [math.CA], 2018.

Crossrefs

Cf. A094638, A130534, A055535 (denominators).

See also A239897/A239898.

Cf. A276977.

Programs

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;

Mathematica

a[n_] := Sum[StirlingS1[n+k, k]/(n+k)!*Sum[(-1)^j/j!, {j, 0, n-k}], {k, 0, n}]; Table[a[n] // Numerator // Abs, {n, 0, 17}] (* Jean-François Alcover, Mar 04 2014, after Maple *)
Numerator[((1-x)^(-1/x)/E + O[x]^20)[[3]]] (* or *)
Numerator[Table[Sum[StirlingS1[n+k, k] Subfactorial[n-k] Binomial[2n, n+k], {k, 0, n}] (-1)^n/(2n)!, {n, 0, 10}]] (* Vladimir Reshetnikov, Sep 23 2016 *)

Formula

See Maple line for formula.

Extensions

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar