A055505 - OEIS

%I #44 Dec 16 2024 12:15:01

%S 1,1,11,7,2447,959,238043,67223,559440199,123377159,29128857391,

%T 5267725147,9447595434410813,1447646915836493,225037938358318573,

%U 29911565062525361,3651003047854884043877,38950782815463986767

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

%C From _Miklos Kristof_, Nov 04 2007: (Start) This is also the sequence of numerators associated with expansion of (1+x)^(1/x).

%C (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 (1+x)^(1/x) = exp(log(1+x)/x) = exp(1)*exp(-x/2)*exp(x^2/3)*exp(x^3/4)*...

%C 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 a(n)/b(n) = Sum_{i>=n} s(i,i-n)/i! where s(n,m) is a Stirling number of the first kind.

%C exp(1) = 1 + Sum_{i>=1} s(i,i)/i!, for the n=1 case.

%C a(1)/b(1) = 1/1 because 1+1/1!+1/2!+1/3!+1/4!+... = exp(1)

%C a(2)/b(2) = 1/2 because 1/2!+3/3!+6/4!+10/5!+... = 1/2*exp(1)

%C a(3)/b(3) = 11/24 because 2/3!+11/4!+35/5!+85/6!+... = 11/24*exp(1)

%C a(4)/b(4) = 7/16 because 6/4!+50/5!+225/6!+735/7!+... = 7/16*exp(1) (End)

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

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

%H G. C. Greubel, <a href="/A055505/b055505.txt">Table of n, a(n) for n = 0..250</a>

%H Markus Brede, <a href="http://dx.doi.org/10.1007/BF02985832">On the convergence of the sequence defining Euler's number</a>, Math. Intelligencer, 27, no. 3 (2005), 6-7.

%H Chao-Ping Chen and Junesang Choi, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.04.338">An Asymptotic Formula for (1+1/x)^x Based on the Partition Function</a>, Amer. Math. Monthly 121 (2014), no. 4, 338--343. MR3183017.

%H Branko Malesevic, Yue Hu, and Cristinel Mortici, <a href="https://arxiv.org/abs/1801.04963">Accurate Estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit</a>, arXiv:1801.04963 [math.CA], 2018.

%F See Maple line for formula.

%e 1+1/2*x+11/24*x^2+7/16*x^3+2447/5760*x^4+...

%e 1, -1/2, 11/24, -7/16, 2447/5760, -959/2304, 238043/580608, -67223/165888, ...

%p 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;

%t 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 *)

%t Numerator[((1-x)^(-1/x)/E + O[x]^20)[[3]]] (* or *)

%t 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 *)

%Y Cf. A094638, A130534, A055535 (denominators).

%Y See also A239897/A239898.

%Y Cf. A276977.

%K nonn,frac

%O 0,3

%A _N. J. A. Sloane_, Jul 11 2000

%E Edited by _N. J. A. Sloane_, Jul 01 2008 at the suggestion of _R. J. Mathar_