A030178 -id:A030178 - cp's OEIS frontend

A126584 Decimal expansion of solution to exp(-x) = x^3.

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

Offset: 0

Denton J. Dailey (denton.dailey(AT)bc3.edu), Jan 05 2007

			0.7728829591492101128487486048782933727290779425096134746...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A030178.

RealDigits[ FindRoot[ Exp[ -x] == x^3, {x, {.5, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]
RealDigits[ 3*ProductLog[1/3], 10, 102] // First (* Jean-François Alcover, Feb 27 2013 *)

3*lambertw(1/3) \\ G. C. Greubel, Mar 06 2018

Equals 3*LambertW(1/3). - G. C. Greubel, Mar 06 2018

A019474 Continued fraction expansion of W(1), where W(x) is the Lambert W function (the root of w*exp(w) = x).

Original entry on oeis.org

0, 1, 1, 3, 4, 2, 10, 4, 1, 1, 1, 1, 2, 7, 306, 1, 5, 1, 2, 1, 5, 1, 1, 1, 1, 7, 1, 4, 2, 15, 1, 2, 1, 1, 4, 1, 3, 3, 5, 4, 1, 1, 1, 4, 3, 1, 38, 1, 2, 4, 1, 5, 2, 1, 6, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 3, 2, 11, 1, 1, 1, 49, 4, 1, 1, 1

Offset: 0

Robert Corless (rmc(AT)pineapple.apmaths.uwo.ca), N. J. A. Sloane

			0.5671432904097838...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for continued fractions for constants

Cf. A030178.

Digite := 80: evalf(LambertW(1)); convert(%,confrac);

ContinuedFraction[ ProductLog[1], 78]  (* Jean-François Alcover, Jun 24 2013 *)

contfrac(lambertw(1)) \\ G. C. Greubel, Mar 03 2018

A115287 Decimal expansion of 1/(1+LambertW(1)).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 19 2006

			0.63810374336511077852...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Cristina B. Corcino, Roberto B. Corcino and István Mező, Continued fraction expansions for the Lambert W function, Aequationes mathematicae, Vol. 93, No. 2 (2019), pp. 485-49.
Mathematics Stack Exchange, Interesting integral related to the Omega Constant/Lambert W Function, 2011.
Victor H. Moll, Some Questions in the Evaluation of Definite Integrals, MAA Short Course, San Antonio, TX, January 2006.
Eric Weisstein's World of Mathematics, Omega Constant.
Eric Weisstein's World of Mathematics, Definite Integral.

Cf. A006153, A030178, A097172, A097173, A097174, A265953, A276231, A302399.

RealDigits[1/(1 + ProductLog[1]), 10, 105] // First (* Jean-François Alcover, Feb 07 2013 *)

1/(1 + lambertw(1)) \\ G. C. Greubel, Nov 05 2017

Equals Integral_{x=-oo..oo} 1/(Pi^2 + (exp(x)-x)^2) dx (discovered by Victor Adamchik). - Amiram Eldar, Jul 04 2021

A200320 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2/2).

Original entry on oeis.org

1, 1, 3, 18, 150, 1590, 20580, 314790, 5554710, 111071520, 2482076520, 61301435580, 1658129152680, 48749053413060, 1547849157554700, 52785934927525800, 1924269399236784600, 74672595203551745400, 3073314600152521124400, 133716009695044269893400, 6132253708189762323370200

Offset: 1

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 18*x^4/4! + 150*x^5/5! +...
where A(1+x - exp(x^2/2)) = x and A(x) = x-1 + exp(A(x)^2/2).

Cf. A200319, A213641, A030178.

Rest[CoefficientList[InverseSeries[Series[1 - E^(x^2/2) + x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=n!*polcoeff(serreverse(1+x-exp(x^2/2+x^2*O(x^n))),n)}

E.g.f.: Series_Reversion(1+x - exp(x^2/2)).

a(n) ~ n^(n-1) * c^(n/2) / (sqrt(1+c) * exp(n) * (c-1+sqrt(c))^(n-1/2)), where c = LambertW(1) = 0.5671432904... (see A030178). - Vaclav Kotesovec, Jan 10 2014

A342359 Decimal expansion of arctan(sqrt(Omega)), where Omega=LambertW(1) is the Omega constant.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

The sine and the cosine of this angle appears in the values of two definite integrals that involve non-principal real branch of the Lambert W function, see A342360 and A342361.

			0.6454752445650039244357315545660663652246772055940215161816...

Cf. A342360, A342361, A030178.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[xi,120]

PARI
```
atan(sqrt(lambertw(1)))
```

Equals arctan(sqrt(LambertW(1))).

A342360 Decimal expansion of 1/(Omega+1)^2, where Omega=LambertW(1) is the Omega constant.

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

			0.40717638729656715790289020473539767731...

Cf. A342359, A342361, A030178, A030797.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[Cos[xi]^4,120]
Omega=LambertW[1]; N[1/(Omega+1)^2,120]
Omega=LambertW[1]; omega=1/Omega; NIntegrate[(-t/LambertW[-1,-t*Omega^omega])^Omega,{t,0,1}, WorkingPrecision->120]

PARI
```
cos(atan(sqrt(lambertw(1))))^4
```
PARI
```
my(Omega=lambertw(1)); 1/(Omega+1)^2
```

Equals cos(A342359)^4 = 1/(A030178+1)^2 = (1-sqrt(A342361))^2.
Equals Integral_{t=0..1} (-t/LambertW(-1,-t*Omega^omega))^Omega, where omega=1/Omega=1/LambertW(1).
Equals A115287^2. - Vaclav Kotesovec, Mar 12 2021

A342361 Decimal expansion of 1/(omega+1)^2, where omega=1/LambertW(1).

Original entry on oeis.org

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

Offset: 0

Gleb Koloskov, Mar 09 2021

			0.1309689005663456008580754336956370484226429615564731843

Cf. A342359, A342360, A030178, A030797.

Omega=LambertW[1]; xi=ArcTan[Sqrt[Omega]]; N[Sin[xi]^4,120]
omega=1/LambertW[1]; N[1/(omega+1)^2,120]
Omega=LambertW[1]; omega=1/Omega; NIntegrate[(-t/LambertW[-1,-t*Omega^omega])^omega,{t,0,1}, WorkingPrecision->120]
RealDigits[1/(1/LambertW[1]+1)^2,10,120][[1]] (* Harvey P. Dale, Mar 26 2025 *)

my(Omega=lambertw(1), xi=atan(sqrt(Omega))); sin(xi)^4

PARI
```
1/(1/lambertw(1)+1)^2
```

Equals Integral_{t=0..1} (-t/W(-1,-t*Omega^omega))^omega, where omega = 1/Omega = 1/LambertW(1).

Equals sin(A342359)^4 = 1/(A030797+1)^2 = (1-sqrt(A342360))^2.

A369550 Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).

Original entry on oeis.org

1, 1, 3, 13, 85, 701, 6901, 79045, 1049385, 15924025, 271248121, 5108389001, 105158055949, 2346022349269, 56348945801877, 1449434215375021, 39758549273200081, 1159092552400164977, 35813081725133941297, 1169791166246561367697, 40297553373717279300981, 1460613225168596836153741

Offset: 0

Paul D. Hanna, Jan 29 2024

nonn

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + 15924025*x^9/9! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
exp(-x) * A(x) = A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 380*x^5/5! + 3750*x^6/6! +  + 42882*x^7/7! + 576296*x^8/8! + ...
The logarithm of e.g.f. A(x) equals L(x) where L(x) = x + L(x^2*exp(x)),
L(x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...

Cf. A369090, A369091, A369551.

{a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1,n, A = exp(X) * subst(A,x,x^2*exp(X)) ); n!*polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = exp(x) * A(x^2*exp(x)).
(2) A(x) = exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = exp(L(x)) where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
(4) A(x) = G(x)/x where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
a(n) = A369090(n+1)/(n+1) for n >= 0.

A369551 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xexp(x) A(x^2*exp(x)).

Original entry on oeis.org

1, 1, 2, 9, 52, 365, 3126, 33607, 434120, 6397785, 104813290, 1881831611, 36703128012, 773468319637, 17544261523166, 427299522260535, 11158470652237456, 311944165793916977, 9313905287778153426, 296051128664550195763, 9979462242106491507860, 355292353569342771519021

Offset: 0

Paul D. Hanna, Jan 29 2024

nonn

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 365*x^5/5! + 3126*x^6/6! + 33607*x^7/7! + 434120*x^8/8! + 6397785*x^9/9! + 104813290*x^10/10! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2550*x^6/6! + 29442*x^7/7! + 386456*x^8/8! + ...
where A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
The expansion of exp(x*A(x)) is the e.g.f. of A369550, which begins
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + ... + A369550(n)*x^n/n! + ...

Cf. A369090, A369091, A369550.

{a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1, n, A = 1 + x*exp(X) * subst(A, x, x^2*exp(X)) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
(2) A(x) = (1/x) * Sum_{n>=0} F(n), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = log(G(x)) / x where G(x) = exp(x) * G(x^2*exp(x)) is the e.g.f. of A369550.
(4) A(x) = L(x)/x where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
a(n) = A369091(n+1)/(n+1) for n >= 0.

A126585 Decimal expansion of solution to exp(-x) = x^4.

Original entry on oeis.org

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

Offset: 0

Denton J. Dailey (denton.dailey(AT)bc3.edu), Jan 05 2007

			0.81555341880896065777272732530855948059740990884063853...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A030178.

RealDigits[ FindRoot[ Exp[ -x] == x^4, {x, {.5, 1}}, WorkingPrecision -> 120][[1, 2, 1]], 10, 111][[1]]
RealDigits[ 4*ProductLog[1/4], 10, 102] // First (* Jean-François Alcover, Feb 27 2013 *)

4*lambertw(1/4) \\ G. C. Greubel, Mar 06 2018

Equals 4*LambertW(1/4). - G. C. Greubel, Mar 06 2018

cp's OEIS Frontend

A126584 Decimal expansion of solution to exp(-x) = x^3.

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A019474 Continued fraction expansion of W(1), where W(x) is the Lambert W function (the root of w*exp(w) = x).

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A115287 Decimal expansion of 1/(1+LambertW(1)).

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A200320 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2/2).

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A342359 Decimal expansion of arctan(sqrt(Omega)), where Omega=LambertW(1) is the Omega constant.

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A342360 Decimal expansion of 1/(Omega+1)^2, where Omega=LambertW(1) is the Omega constant.

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A342361 Decimal expansion of 1/(omega+1)^2, where omega=1/LambertW(1).

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A369550 Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).

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A369551 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xexp(x) A(x^2*exp(x)).