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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 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: A002743 A220773 A290772 * 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

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Last modified December 15 04:33 EST 2018. Contains 318141 sequences. (Running on oeis4.)