A030178 -id:A030178 - cp's OEIS frontend

A227457 E.g.f. equals the series reversion of x - x*log(1+x).

1, 2, 9, 68, 720, 9804, 163184, 3210192, 72870120, 1874721360, 53905894152, 1713195438624, 59633476003920, 2256257009704320, 92196226214092800, 4046446853549201664, 189845257963376620800, 9481546020840245199360, 502242773970728703225600, 28124368575613839072714240

Offset: 1

Paul D. Hanna, Jul 12 2013

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 68*x^4/4! + 720*x^5/5! +...
where A(x) = x/(1 - log(1+A(x))).
The e.g.f. satisfies:
(3) A(x) = x + x*log(1+x) + d/dx x^2*log(1+x)^2/2! + d^2/dx^2 x^3*log(1+x)^3/3! + d^3/dx^3 x^4*log(1+x)^4/4! +...
(4) log(A(x)/x) = log(1+x) + d/dx x*log(1+x)^2/2! + d^2/dx^2 x^2*log(1+x)^3/3! + d^3/dx^3 x^3*log(1+x)^4/4! +...

Cf. A052819, A030178.

Rest[CoefficientList[InverseSeries[Series[x - x*Log[1+x],{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=n!*polcoeff(serreverse(x-x*log(1+x +x*O(x^n))), n)}
for(n=1,25,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*log(1+x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*log(1+x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{a(n)=sum(k=0,n-1,k!*Stirling1(n-1,k)*binomial(n+k-1,n-1))}
for(n=1,25,print1(a(n),", "))

a(n) = Sum_{k=0..n-1} k! * Stirling1(n-1,k) * binomial(n+k-1,n-1). [From a formula in A052819 due to Vladimir Kruchinin]
E.g.f. A(x) satisfies:
(1) A(x - x*log(1+x)) = x.
(2) A(x) = x/(1 - log(1+A(x))).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * log(1+x)^n / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * log(1+x)^n / n! ).
a(n) ~ n^(n-1) * (1-c) / (c*sqrt(1+c) * exp(n) * (c-2+1/c)^n), where c = LambertW(1) = 0.5671432904... (see A030178). - Vaclav Kotesovec, Jan 10 2014

A258114 E.g.f.: Sum_{n>=0} x^n * cosh(n*x).

Original entry on oeis.org

1, 1, 2, 9, 72, 665, 6960, 85057, 1199744, 19070865, 336372480, 6522635801, 137996694528, 3163206890857, 78085740701696, 2065239729737745, 58263449436979200, 1746433243580269217, 55428341343200280576, 1856918215298125692073, 65483209810866254643200, 2424691204935999655757241

Offset: 0

Paul D. Hanna, May 20 2015

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 72*x^4/4! + 665*x^5/5! +...
where A(x) = 1 + x*cosh(x) + x^2*cosh(2*x) + x^3*cosh(3*x) + x^4*cosh(4*x) +...

G. C. Greubel, Table of n, a(n) for n = 0..410

Cf. A030178.

CoefficientList[Series[(1-x*Cosh[x])/(1-2*x*Cosh[x]+x^2), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 21 2015 *)

{a(n) = sum(k=0,n, n!/k! * ((n-k)^k + (-n+k)^k)/2)}
for(n=0,30,print1(a(n),", "))

{a(n) = local(A=1); A = sum(m=0,n, x^m*cosh(m*x +x*O(x^n))); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = local(X=x+x*O(x^n),A=1); A = (1 - x*cosh(X)) / (1 - 2*x*cosh(X) + x^2); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: (1 - x*cosh(x)) / (1 - 2*x*cosh(x) + x^2).
a(n) = Sum_{k=0..n} n!/k! * ((n-k)^k + (-n+k)^k)/2.
a(n) ~ n! * (1-c*cosh(c)) / (2*(cosh(c)+c*(sinh(c)-1)) * c^(n+1)), where c = A030178 = LambertW(1) = 0.56714329040978387299996866... . - Vaclav Kotesovec, May 21 2015

A333761 Decimal expansion of root of the equation LambertW(r) = 1 - r.

Original entry on oeis.org

5, 9, 8, 9, 4, 1, 8, 6, 2, 4, 5, 8, 4, 5, 2, 9, 6, 4, 3, 4, 9, 3, 7, 4, 6, 2, 4, 9, 9, 3, 5, 4, 3, 3, 7, 0, 9, 0, 4, 3, 9, 3, 0, 1, 3, 4, 9, 5, 4, 0, 2, 2, 2, 3, 6, 3, 0, 4, 0, 3, 5, 0, 7, 9, 2, 2, 1, 3, 0, 3, 6, 0, 0, 4, 5, 4, 2, 0, 3, 0, 0, 0, 4, 6, 6, 7, 4, 1, 8, 2, 8, 7, 0, 9, 1, 3, 7, 2, 3, 2, 5, 5, 5, 6, 9, 0

Offset: 0

Vaclav Kotesovec, Jun 09 2020

			0.59894186245845296434937462499354337090439301349540222363040350792213...

Eric Weisstein's World of Mathematics, Lambert W-Function.
Wikipedia, Lambert W function.

Cf. A030178, A334242.

RealDigits[r/.FindRoot[LambertW[r] == 1 - r, {r, 1/2}, WorkingPrecision->150], 10, 120][[1]]

solve(x=0, 1, 1-x-lambertw(x)) \\ Michel Marcus, Jun 09 2020

A357549 a(n) = floor( Sum_{k=0..n-1} n^k / (k! * a(k)) ), for n > 0 with a(0) = 1.

Original entry on oeis.org

1, 1, 3, 5, 9, 17, 30, 52, 91, 161, 285, 503, 889, 1573, 2782, 4920, 8697, 15368, 27146, 47928, 84590, 149246, 263247, 464214, 818445, 1442762, 2543025, 4482001, 7898979, 13920609, 24532535, 43234510, 76195273, 134288583, 236682848, 417170144, 735325596, 1296184444

Offset: 0

Paul D. Hanna, Dec 01 2022

nonn

Limit_{n->oo} a(n)/a(n+1) = w = exp(-w) = LambertW(1), the omega constant A030178.

			a(n) = floor(1 + n/a(1) + n^2/(2!*a(2)) + n^3/(3!*a(3)) + n^4/(4!*a(4)) + n^5/(5!*a(5)) + ... + n^(n-1)/((n-1)!*a(n-1)) ), for n > 0 with a(0) = 1.
To generate this sequence, start with a(0) = 1 and proceed as follows:
a(1) = 1;
a(2) = 1 + 2;
a(3) = floor(1 + 3 + 3^2/(2!*3)) = 5;
a(4) = floor(1 + 4 + 4^2/(2!*3) + 4^3/(3!*5)) = 9;
a(5) = floor(1 + 5 + 5^2/(2!*3) + 5^3/(3!*5) + 5^4/(4!*9)) = 17;
a(6) = floor(1 + 6 + 6^2/(2!*3) + 6^3/(3!*5) + 6^4/(4!*9) + 6^5/(5!*17)) = 30;
a(7) = floor(1 + 7 + 7^2/(2!*3) + 7^3/(3!*5) + 7^4/(4!*9) + 7^5/(5!*17) + 7^6/(6!*30)) = 52;
a(8) = floor(1 + 8 + 8^2/(2!*3) + 8^3/(3!*5) + 8^4/(4!*9) + 8^5/(5!*17) + 8^6/(6!*30) + 8^7/(7!*52)) = 91;
...
The terms of this sequence are computed from partial sums; the actual infinite sums: Sum_{k>=0} n^k / (k!*a(k)), for n >= 1, begin:
n = 1: 2.205170228313619257204573175905229637440183827382...
n = 2: 4.026624683096007253196633437972996492234406960420...
n = 3: 6.938404847258827610039050722524656436473915836809...
n = 4: 11.76290965545838695557108226269004580813840600527...
n = 5: 19.93268682960501544009268973006846510258954225008...
n = 6: 33.95355685301572322838214122801051011301028947272...
n = 7: 58.22316762392820953863455561301453509123241732275...
n = 8: 100.4764040611128933206396099594217599817997316217...
n = 9: 174.3356399991557294349025383486302219269780824259...
n = 10: 303.8074912728852469034815183896362125031997652232...
...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A030178.

/* Print a(n) for n = 0 through N */
N = 40; A=vector(N+1);
{ a(n) = if(n<0,0, A[n+1] = if(n<1,1, floor( sum(k=0,n-1, n^k/k!/A[k+1]) ) )) }
for(n=0,N,print1(a(n),", "))

A387101 Decimal expansion of the smallest real solution to e^x = x^3.

Original entry on oeis.org

1, 8, 5, 7, 1, 8, 3, 8, 6, 0, 2, 0, 7, 8, 3, 5, 3, 3, 6, 4, 5, 6, 9, 8, 0, 9, 8, 2, 0, 6, 2, 7, 6, 6, 6, 9, 9, 9, 0, 4, 4, 1, 5, 3, 3, 1, 7, 8, 8, 9, 0, 8, 0, 5, 7, 3, 7, 9, 5, 0, 8, 2, 2, 3, 4, 7, 8, 8, 1, 2, 0, 5, 6, 7, 6, 5, 2, 6, 6, 2, 0, 3, 8, 3, 3, 1, 4, 0, 7, 7, 8, 5, 6, 4, 7, 7, 6, 5, 2, 1

Offset: 1

Stefano Spezia, Aug 16 2025

Equivalently, the smallest real solution to log(x) = x/3.

			1.85718386020783533645698098206276669990441533...

Index entries for transcendental numbers.

Cf. A030178, A126583, A126584, A387102 (largest).

RealDigits[-3*ProductLog[-1/3],10,100][[1]]
RealDigits[x/.FindRoot[E^x==x^3,{x,1},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Sep 02 2025 *)

-3*lambertw(-1/3) \\ Michel Marcus, Aug 18 2025

Equals -3*LambertW(-1/3).

A387102 Decimal expansion of the largest real solution to e^x = x^3.

Original entry on oeis.org

4, 5, 3, 6, 4, 0, 3, 6, 5, 4, 9, 7, 3, 5, 2, 7, 4, 2, 1, 6, 9, 0, 2, 1, 9, 0, 3, 4, 2, 1, 6, 1, 1, 6, 1, 1, 3, 8, 1, 0, 9, 5, 1, 1, 5, 5, 4, 0, 6, 0, 7, 8, 3, 6, 2, 7, 7, 7, 8, 5, 6, 4, 2, 1, 9, 9, 6, 9, 3, 1, 4, 8, 1, 5, 3, 4, 7, 2, 4, 4, 8, 1, 9, 7, 2, 7, 9, 6, 4, 6, 4, 4, 5, 8, 3, 3, 9, 5, 0, 5

Offset: 1

Stefano Spezia, Aug 16 2025

Equivalently, the largest real solution to log(x) = x/3.

			4.5364036549735274216902190342161161138109511554...

Index entries for transcendental numbers.

Cf. A030178, A126583, A126584, A366565, A387101 (smallest).

RealDigits[-3*ProductLog[-1,-1/3],10,100][[1]]

Equals -3*LambertW(-1, -1/3).

A139338 Least k > 0 such that Sum_{i=0..2*n-1} (-k)^i/i! < 0.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 41, 42, 43

Offset: 1

Benoit Cloitre, Jun 08 2008

nonn

Serge Francinou, Herve Gianella and Serge Nicolas, Exercices de mathématiques : oraux X-ENS, Analyse 1, Cassini Editeur, 2003, pp. 119-121.

Cf. A030178.

a(n)=if(n<0,0,k=1;while(sum(i=0,2*n-1,(-k)^i*1./i!)>0,k++);k)

a(n) = r*n + o(n) where r is the solution to x + log(x) = 0 and 0 < x < 1: r = 0.56714329040978387299996... (see A030178).

A246823 Decimal expansion of the asymptotic cost of the minimum edge cover in a complete bipartite graph with independent exponentially distributed edge costs.

Original entry on oeis.org

1, 4, 5, 5, 9, 3, 8, 0, 9, 2, 6, 7, 6, 4, 0, 4, 1, 9, 4, 0, 1, 2, 1, 4, 8, 9, 4, 0, 9, 5, 7, 3, 2, 4, 9, 8, 2, 1, 8, 0, 2, 8, 3, 6, 0, 2, 3, 2, 3, 2, 0, 4, 8, 3, 0, 6, 5, 3, 5, 6, 0, 9, 9, 3, 1, 9, 6, 7, 7, 7, 6, 6, 5, 2, 3, 7, 6, 8, 1, 6, 5, 7, 5, 9, 8, 8, 7, 4, 2, 0, 2, 8, 8, 5, 9, 5, 7, 9, 9, 6, 5, 7, 4, 1

Offset: 1

Jean-François Alcover, Sep 04 2014

			1.4559380926764041940121489409573249821802836023232...

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 55.
Martin Hessler and Johan Wästlund, Edge cover and polymatroid flow problems

Cf. A030178.

RealDigits[ProductLog[1]^2 + 2*ProductLog[1], 10, 104] // First

W(1)^2 + 2W(1), where W is the Lambert W-function (also known as ProductLog).

A265131 Decimal expansion of positive x satisfying x^(x^x) = LambertW(1).

Original entry on oeis.org

4, 4, 3, 3, 4, 4, 8, 8, 7, 3, 5, 7, 9, 1, 5, 0, 7, 4, 1, 5, 9, 8, 0, 0, 2, 7, 9, 3, 7, 8, 8, 6, 8, 8, 6, 0, 1, 2, 2, 5, 4, 1, 3, 9, 6, 5, 2, 2, 2, 2, 9, 2, 1, 4, 9, 5, 7, 7, 1, 3, 5, 9, 5, 4, 0, 8, 8, 4, 9, 4, 5, 4, 8, 8, 1, 8, 6, 0, 0, 2, 4, 6, 5, 9, 7, 8, 8, 6, 7, 6, 8, 7, 9, 2, 2, 8, 4, 9, 2, 5, 1, 9, 9, 4, 1, 5, 3, 0, 0, 1, 1, 9, 8, 1

Offset: 0

Anders Hellström, Dec 02 2015

			0.44334488735791507415980027937886886012254139652223...

Wikipedia, Lambert W function
Wikipedia, Omega constant

Cf. A030178, A264807, A264808.

RealDigits[x/.FindRoot[x^(x^x)==ProductLog[1],{x,1},WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Jul 19 2020 *)

default(realprecision,2000);solve(x=0.001,3,x^(x^x)-lambertw(1))

A266092 Decimal expansion of the power tower of 1/sqrt(3): the real solution to 3^(x/2)*x = 1.

Original entry on oeis.org

6, 8, 6, 0, 2, 6, 7, 2, 4, 5, 3, 6, 2, 5, 1, 3, 1, 9, 7, 1, 3, 0, 0, 6, 8, 4, 6, 1, 8, 2, 2, 3, 8, 1, 5, 9, 5, 0, 3, 3, 2, 4, 2, 3, 7, 7, 6, 2, 3, 4, 3, 4, 0, 2, 4, 1, 7, 6, 7, 1, 9, 1, 6, 7, 0, 0, 4, 0, 2, 9, 0, 5, 8, 1, 8, 7, 5, 4, 8, 4, 8, 7, 7, 6, 4, 2, 8, 1, 5, 7, 8, 6, 8, 9, 3, 9, 8, 2, 6, 3, 8, 0, 6, 6, 8, 6, 9, 9, 3, 5, 2, 8, 3, 3, 2, 4, 8, 9, 6, 7

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

			(1/sqrt(3))^(1/sqrt(3))^(1/sqrt(3))^(1/sqrt(3))^… = 0.686026724536251319713006846182…

Eric Weisstein's World of Mathematics, Power Tower
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A020760, A030178, A073084, A073243, A231096.

RealDigits[(2 ProductLog[Log[3]/2])/Log[3], 10, 120][[1]]

t=log(3)/2; lambertw(t)/t \\ Charles R Greathouse IV, Apr 18 2016

Equals 2*LambertW(log(3)/2)/log(3).

cp's OEIS Frontend

A227457 E.g.f. equals the series reversion of x - x*log(1+x).

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A258114 E.g.f.: Sum_{n>=0} x^n * cosh(n*x).

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A333761 Decimal expansion of root of the equation LambertW(r) = 1 - r.

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A357549 a(n) = floor( Sum_{k=0..n-1} n^k / (k! * a(k)) ), for n > 0 with a(0) = 1.

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A387101 Decimal expansion of the smallest real solution to e^x = x^3.

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A387102 Decimal expansion of the largest real solution to e^x = x^3.

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A139338 Least k > 0 such that Sum_{i=0..2*n-1} (-k)^i/i! < 0.

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A246823 Decimal expansion of the asymptotic cost of the minimum edge cover in a complete bipartite graph with independent exponentially distributed edge costs.