A055535 -id:A055535 - cp's OEIS frontend

A239898 Bisection of A055535.

1, 24, 5760, 580608, 1393459200, 73574645760, 24103053950976000, 578473294823424000, 9440684171518279680000, 271211974879377138647040000, 3579998068407778230140928000000, 282308419108727654719684608000000, 258955866680053703121272297226240000000

Offset: 0

N. J. A. Sloane, Apr 05 2014

Based on a very good approximation to e.

			Denominators of the fractions 1, 11/24, 2447/5760, 238043/580608, ... (see A055505/A055535).

Harlan J. Brothers and John A. Knox,, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, 20 (1998), 25-29. MR1646709 (2000c:11209).
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.
John A. Knox and Harlan J. Brothers, Novel series-based approximations to e, College Math. J. 30 (1999), no. 4, 269-275. MR1717867 (2000i:11198).

Cf. A055505/A055535, A239897 (numerators).

More terms from Amiram Eldar, May 08 2024

A055505 Numerators in expansion of (1-x)^(-1/x)/e.

Original entry on oeis.org

1, 1, 11, 7, 2447, 959, 238043, 67223, 559440199, 123377159, 29128857391, 5267725147, 9447595434410813, 1447646915836493, 225037938358318573, 29911565062525361, 3651003047854884043877, 38950782815463986767

Offset: 0

N. J. A. Sloane, Jul 11 2000

From Miklos Kristof, Nov 04 2007: (Start) This is also the sequence of numerators associated with expansion of (1+x)^(1/x).
(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 - ...).
(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 this sequence, let b(n) be A055535. 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)

			1+1/2*x+11/24*x^2+7/16*x^3+2447/5760*x^4+...
1, -1/2, 11/24, -7/16, 2447/5760, -959/2304, 238043/580608, -67223/165888, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.1.

G. C. Greubel, Table of n, a(n) for n = 0..250
Markus Brede, On the convergence of the sequence defining Euler's number, Math. Intelligencer, 27, no. 3 (2005), 6-7.
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.
Branko Malesevic, Yue Hu, and Cristinel Mortici, Accurate Estimates of (1+x)^{1/x} Involved in Carleman Inequality and Keller Limit, arXiv:1801.04963 [math.CA], 2018.

Cf. A094638, A130534, A055535 (denominators).
See also A239897/A239898.
Cf. A276977.

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;

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

See Maple line for formula.

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A276977 Duplicate of A106827.

Original entry on oeis.org

1, 1, 11, 315, 17129, 1510425, 196385475, 35327367075, 8399994587985, 2550903574364145, 963207568455370875, 442613044315692124875, 243195136160954426677305, 157442856285298191126143625, 118607799383105394973766029875, 102867257381973743111023517821875

Offset: 0

Vladimir Reshetnikov, Sep 23 2016

dead

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A055505, A055535, A000166.

Table[(-1)^n Sum[StirlingS1[n+k, k] Subfactorial[n-k] Binomial[2n, n+k], {k, 0, n}], {n, 0, 20}]

a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n+k, k) * !(n-k) * C(2*n, n+k), where !n = A000166(n) is the subfactorial, C(n,k) are binomial coefficients.

A106827 Numerators in expansion of (1 - x)^(-1/x) / e.

Original entry on oeis.org

1, 1, 11, 315, 17129, 1510425, 196385475, 35327367075, 8399994587985, 2550903574364145, 963207568455370875, 442613044315692124875, 243195136160954426677305, 157442856285298191126143625, 118607799383105394973766029875, 102867257381973743111023517821875

Offset: 0

Philippe Deléham, May 21 2005

			G.f. = 1 + 1*x/2! + 11*x^2/4! + 315*x^3/6! + 17129*x^4/8! + 503475*x^5/10! + ...

L. Comtet, Analyse Combinatoire, P. U. F., 1970, tome second, p. 140, #12.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, Problem 11.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1 . 3 . 1.

G. C. Greubel, Table of n, a(n) for n = 0..201

Cf. A055505, A055535.

m:=31; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&*[Exp(x^(j-1)/j): j in [2..40]]) )); [Factorial(2*n-2)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 14 2021

Table[(-1)^n Sum[StirlingS1[n+k, k] Subfactorial[n-k] Binomial[2n, n+k], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 23 2016 *)
With[{m=30}, CoefficientList[Series[(1-x)^(-1/x)/E, {x,0,m}], x]*(2*Range[0,m])!] (* G. C. Greubel, Sep 14 2021 *)

def A_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( product(exp(x^(j-1)/j) for j in (2..41)) ).list()
A=A_list(40)
[factorial(2*n)*A[n] for n in (0..31)] # G. C. Greubel, Sep 14 2021

Sum_{n>=0} a(n)/(2n)!*x^n = (1 - x)^(-1/x) / e.
a(n) = A055505(n)*(2n)! / A055535(n).
a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n+k, k) * !(n-k) * C(2*n, n+k), where !n = A000166(n) is the subfactorial, C(n,k) are binomial coefficients. - Vladimir Reshetnikov, Sep 23 2016
a(n) = (2*n)! * coefficients of Product_{j >= 2} exp(x^(j-1)/j). - G. C. Greubel, Sep 14 2021

a(5) corrected by G. C. Greubel, Sep 14 2021

A239897 Bisection of A055505.

Original entry on oeis.org

1, 11, 2447, 238043, 559440199, 29128857391, 9447595434410813, 225037938358318573, 3651003047854884043877, 104388909491649724435759747, 1372557084260440289321615059133, 107881945709178295095123859185817, 98682616643700175634367947900986085893

Offset: 0

N. J. A. Sloane, Apr 05 2014

Based on a very good approximation to e.

			Numerators of the fractions 1, 11/24, 2447/5760, 238043/580608, ... (see A055505/A055535).

Harlan J. Brothers and John A. Knox,, New closed-form approximations to the logarithmic constant e, Math. Intelligencer, 20 (1998), 25-29. MR1646709 (2000c:11209).
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.
John A. Knox and Harlan J. Brothers, Novel series-based approximations to e, College Math. J. 30 (1999), no. 4, 269-275. MR1717867 (2000i:11198).

Cf. A055505/A055535, A239898 (denominators).

More terms from Amiram Eldar, May 08 2024

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A239898 Bisection of A055535.

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A055505 Numerators in expansion of (1-x)^(-1/x)/e.

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A276977 Duplicate of A106827.

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A106827 Numerators in expansion of (1 - x)^(-1/x) / e.

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A239897 Bisection of A055505.

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