A055535 Denominators in expansion of (1-x)^(-1/x)/e.

1, 2, 24, 16, 5760, 2304, 580608, 165888, 1393459200, 309657600, 73574645760, 13377208320, 24103053950976000, 3708162146304000, 578473294823424000, 77129772643123200, 9440684171518279680000, 100969884187361280000

Offset

0,2

Comments

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

References

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

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

Links

Robert Israel, Table of n, a(n) for n = 0..377

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.

Formula

From Miklos Kristof, 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_{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)

Example

(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 + ...).

Maple

G:= (1-x)^(-1/x)/exp(1):
S:= series(G, x, 32):
seq(denom(coeff(S, x, j)), j=0..30); # Robert Israel, Sep 23 2016

Mathematica

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

Crossrefs

Cf. A094638, A130534, A055505 (numerators), A276977.

Sequence in context: A220773 A290772 A342545 * A072217 A229429 A281139

Adjacent sequences: A055532 A055533 A055534 * A055536 A055537 A055538

Keyword

nonn,frac

Author

N. J. A. Sloane, Jul 11 2000

Extensions

Edited by N. J. A. Sloane, Jul 25 2008 at the suggestion of R. J. Mathar and Eric Rowland

Status

approved