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A055535 Denominators in expansion of (1-x)^(-1/x)/e. 6
1, 2, 24, 16, 5760, 2304, 580608, 165888, 1393459200, 309657600, 73574645760, 13377208320, 24103053950976000, 3708162146304000, 578473294823424000, 77129772643123200, 9440684171518279680000, 100969884187361280000 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)