A055535 - OEIS

%I #34 Dec 16 2024 13:28:07

%S 1,2,24,16,5760,2304,580608,165888,1393459200,309657600,73574645760,

%T 13377208320,24103053950976000,3708162146304000,578473294823424000,

%U 77129772643123200,9440684171518279680000,100969884187361280000

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

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

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

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

%H Robert Israel, <a href="/A055535/b055535.txt">Table of n, a(n) for n = 0..377</a>

%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.

%F From _Miklos Kristof_, Nov 04 2007 (Start):

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

%F Let a(n) be A055505, let b(n) be this sequence. Then (1+x)^(1/x) = exp(1)*a(n)/b(n) x^n.

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

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

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

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

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

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

%e (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 + ...).

%p G:= (1-x)^(-1/x)/exp(1):

%p S:= series(G,x,32):

%p seq(denom(coeff(S,x,j)),j=0..30); # _Robert Israel_, Sep 23 2016

%t 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 *)

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

%t 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 *)

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

%K nonn,frac

%O 0,2

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

%E Edited by _N. J. A. Sloane_, Jul 25 2008 at the suggestion of _R. J. Mathar_ and _Eric Rowland_