A030178 -id:A030178 - cp's OEIS frontend

A277682 Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Imaginary part of the complex constant z_3 whose real part is in A277681 (see the latter entry for more information).

			7.588631178472512622568923954107584383013473671992856360409437...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681 (real part), A277683 (modulus).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

RealDigits[Im[ProductLog[1, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

Let z_3 = A277681+i*A277682. Then z_3 = exp(z_3) = log(z_3)+2*Pi*i = -W_-2(-1).

A277683 Decimal expansion of the modulus of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 12 2016

Modulus of z_3 = A277681 + i*A277682. See A277681 for more information.

			7.863861176094232668842573623487382321468320207779893460294144...

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Fixed points of +exp(z): z_1: A059526, A059527, A238274, and z_3: A277681 (real part), A277682 (imaginary part).

Fixed points of -exp(z): z_0: A030178, and z_2: A276759, A276760, A276761.

RealDigits[Norm[ProductLog[1, -1]], 10, 105][[1]] (* Jean-François Alcover, Nov 12 2016 *)

default(realprecision,2050);eps=5.0*10^(default(realprecision))
M(z,K)=log(z)+2*Pi*K*I; \\ the convergent mapping (any K)
K=1;z=1+I;zlast=z;
while(1,z=M(z,K);if(abs(z-zlast)

A126583 Decimal expansion of solution to exp(-x) = x^2.

Original entry on oeis.org

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

Offset: 0

Denton J. Dailey (djd1497(AT)aol.com), Jan 05 2007

The value of the infinite power tower function x^x^x... at x = sqrt(1/e). - Alois P. Heinz, Oct 19 2016

			0.7034674224983916520498186018599021303429284310342236...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration
Index entries for transcendental numbers

Cf. A030178, A099554, A202356.

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

2*lambertw(1/2) \\ G. C. Greubel, Mar 06 2018

Equals 2*LambertW(1/2). - Alois P. Heinz, Oct 19 2016

Equals log(A099554) = 2*A202356. - Hugo Pfoertner, Jul 19 2024

A198860 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x))).

Original entry on oeis.org

1, 1, 3, 17, 144, 1634, 23312, 401274, 8096680, 187472136, 4900535832, 142766286552, 4587190461840, 161161214978880, 6146415080939520, 252902928346825104, 11167368115492742400, 526752556713346955520, 26433830208985721222400, 1406218428780691953635712

Offset: 0

Paul D. Hanna, Oct 30 2011

nonn

Compare to e.g.f. G(x) of A052802, which satisfies: G(x) = 1/(1 + log(1 - x*G(x))).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 144*x^4/4! + 1634*x^5/5! + ...
where log(1 + x*A(x)) equals
1 - 1/A(x) = x + x^2/2! + 5*x^3/3! + 38*x^4/4! + 404*x^5/5! + 5514*x^6/6! + ...

Cf. A030178, A052802.

a[n_] := Sum[ Binomial[n+k, n]*Sum[ (-1)^(j)*Binomial[k, j]*Sum[ (-1)^i*i!*Binomial[j, i]*StirlingS1[n, i], {i, 0, j}], {j, 0, k}], {k, 0, n}]/(n+1); Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Jun 24 2013, after Vladimir Kruchinin *)
CoefficientList[1/x*InverseSeries[Series[x-x*Log[1+x], {x, 0, 20}], x],x] * Range[0, 19]! (* Vaclav Kotesovec, Dec 28 2013 *)

a(n):=sum(binomial(n+k,n)*sum((-1)^(j)*binomial(k,j)*sum((-1)^i*i!*binomial(j,i)* stirling1(n,i),i,0,j),j,0,k),k,0,n)/(n+1); /* Vladimir Kruchinin, Feb 04 2012 */

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

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

/* by Vladimir Kruchinin's formula: */
{a(n)=sum(k=0,n,binomial(n+k,n)*sum(j=0,k,(-1)^(j)*binomial(k,j)*sum(i=0,j,(-1)^i*i!*binomial(j,i)*stirling(n,i,1))))/(n+1)}

E.g.f. satisfies: A(x*(1 - log(1+x))) = 1/(1 - log(1+x)).
E.g.f.: A(x) = (1/x)*Series_Reversion[x - x*log(1+x)].
a(n) = n!*[x^n] 1/(1 - log(1+x))^(n+1)/(n+1).
a(n) = Sum_{k=0..n} (binomial(n+k,n) * Sum_{j=0..k} (-1)^(j)*binomial(k,j) * (Sum_{i=0..j} (-1)^i*i!*binomial(j,i)*Stirling1(n,i)))/(n+1). - Vladimir Kruchinin, Feb 04 2012
a(n) ~ n^(n-1) / ((1-c)*sqrt(1+c) * exp(n) * (1/c+c-2)^n), where c = LambertW(1). - Vaclav Kotesovec, Dec 28 2013
a(n) = (1/(n+1)!) * Sum_{k=0..n} (n+k)! * Stirling1(n,k). - Seiichi Manyama, Nov 06 2023

A245265 E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).

Original entry on oeis.org

1, 1, 3, 37, 649, 15461, 471571, 17456041, 760880625, 38178439849, 2167446089251, 137359883836781, 9612722107574521, 736277501363180557, 61265207586681046131, 5503291392884323494961, 530778414439201798454881, 54706967800114521799571921, 6000952913613549583603208515

Offset: 0

Vaclav Kotesovec, Jul 15 2014

Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Cf. A161630 (p=1), A212722 (p=2), A212917 (p=3).
Cf. A030178.
Cf. A366234 (log).

Table[Sum[n! * (1 + 4*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}]

for(n=0,30, print1(sum(k=0,n, n!*(1 + 4*(n-k))^(k-1)/k!*binomial(n-1,n-k)), ", ")) \\ G. C. Greubel, Nov 17 2017

a(n) = Sum_{k=0..n} n! * (1 + 4*(n-k))^(k-1)/k! * C(n-1,n-k).

a(n) ~ n^(n-1) * (1+2*LambertW(1))^(n+1/2) / (exp(n) * (LambertW(1))^(2*n) * (4*sqrt(1+LambertW(1)))). - Vaclav Kotesovec, Jul 15 2014

A369090 Expansion of e.g.f. A(x) satisfying A(x) = A( x^2*exp(x) ) / x, with A(0) = 0.

Original entry on oeis.org

1, 2, 9, 52, 425, 4206, 48307, 632360, 9444465, 159240250, 2983729331, 61300668012, 1367054727337, 32844312889766, 845234187028155, 23190947446000336, 675895337644401377, 20863665943202969586, 680448552777544884643, 23395823324931227353940, 846248620848062865320601

Offset: 1

Paul D. Hanna, Jan 26 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) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! + 4206*x^6/6! + 48307*x^7/7! + 632360*x^8/8! + 9444465*x^9/9! + 159240250*x^10/10! + ...
RELATED SERIES.
The expansion of the logarithm of A(x)/x starts
log(A(x)/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! + ...
and equals the sum of all iterations of the function x^2*exp(x).
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2/2! + 3*x^3/3! + 8*x^4/4! - 155*x^5/5! + 1464*x^6/6! - 7931*x^7/7! - 65360*x^8/8! + 2742345*x^9/9! + ...
then R(x) and e.g.f. A(x) satisfy:
(1) R( A(x) ) = x,
(2) R( x*A(x) ) = x^2 * exp(x).
GENERATING METHOD.
Let F(n) equal the n-th iteration of x^2*exp(x), so that
F(0) = x,
F(1) = x^2 * exp(x),
F(2) = x^4 * exp(2*x) * exp(x^2*exp(x)),
F(3) = x^8 * exp(4*x) * exp(2*x^2*exp(x)) * exp(F(2)),
F(4) = x^16 * exp(8*x) * exp(4*x^2*exp(x)) * exp(2*F(2)) * exp(F(3)),
F(5) = x^32 * exp(16*x) * exp(8*x^2*exp(x)) * exp(4*F(2)) * exp(2*F(3)) * exp(F(4)),
...
F(n+1) = F(n)^2 * exp(F(n))
...
Then the e.g.f. A(x) equals
A(x) = x * exp(F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...).
equivalently,
A(x) = x * exp(x + x^2*exp(x) + x^4*exp(2*x)*exp(x^2*exp(x)) + x^8*exp(4*x)*exp(2*x^2*exp(x)) * exp(x^4*exp(2*x)*exp(x^2*exp(x))) + ...).

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A369091, A369550 (a(n)/n), A030178.

Cf. A367390.

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

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

A299617 Decimal expansion of e^(W(1) + W(e)) = e/(W(1)*W(e)), where W is the Lambert W function (or PowerLog); see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 01 2018

The Lambert W function satisfies the functional equation e^(W(x) + W(y)) = x*y/(W(x)*W(y)) for x and y greater than -1/e, so that e^(W(1) + W(e)) = e/(W(1)*W(e)). See A299613 for a guide to related constants.

			e^(W(1) + W(e)) = 4.7929365901428140257258473723821086015...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A299613, A030178.

w[x_] := ProductLog[x]; x = 1; y = E;
N[E^(w[x] + w[y]), 130]   (* A299617 *)
RealDigits[E/(LambertW[1]*LambertW[E]), 10, 100][[1]] (* G. C. Greubel, Mar 03 2018 *)

exp(1)/(lambertw(1)*lambertw(exp(1))) \\ G. C. Greubel, Mar 03 2018

A300916 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/e.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Mar 16 2018

			0.557919282255416046773864733137284325268059522147000568856861678665669168...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A001113, A030178, A068985, A277522, A277651, A293009.

RealDigits[E*Exp[-2*LambertW[1]]/(1+LambertW[1]), 10, 100][[1]] (* G. C. Greubel, Sep 09 2018 *)

exp(1)*exp(-2*lambertw(1))/(1+lambertw(1)) \\ Michel Marcus, Mar 16 2018

Equals exp(1)*exp(-2*LambertW(1))/(1+LambertW(1)).

A369091 Expansion of e.g.f. A(x) satisfying A(x) = x + A( x^2*exp(x) ), with A(0) = 0.

Original entry on oeis.org

1, 2, 6, 36, 260, 2190, 21882, 268856, 3907080, 63977850, 1152946190, 22581979332, 477140664156, 10828556474918, 263163922847490, 6836792356168560, 189694001088036752, 5614994984290505586, 176964200467784915094, 5921022573291003915260, 209568707084236321665060

Offset: 1

Paul D. Hanna, Jan 26 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) = 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! + 3907080*x^9/9! + 63977850*x^10/10! + ...
which equals the sum of all iterations of the function x^2*exp(x).
RELATED SERIES.
x*exp(A(x)) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! + 4206*x^6/6! + 48307*x^7/7! + 632360*x^8/8! + ... + A369090(n)*x^n/n! + ...
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2/2! + 6*x^3/3! - 36*x^4/4! + 340*x^5/5! - 3870*x^6/6! + 52038*x^7/7! - 850472*x^8/8! + 16378920*x^9/9! + ...
then R(x) and e.g.f. A(x) satisfy:
(1) R( A(x) ) = x,
(2) R( A(x) - x ) = x^2 * exp(x).
GENERATING METHOD.
Let F(n) equal the n-th iteration of x^2*exp(x), so that
F(0) = x,
F(1) = x^2 * exp(x),
F(2) = x^4 * exp(2*x) * exp(x^2*exp(x)),
F(3) = x^8 * exp(4*x) * exp(2*x^2*exp(x)) * exp(F(2)),
F(4) = x^16 * exp(8*x) * exp(4*x^2*exp(x)) * exp(2*F(2)) * exp(F(3)),
F(5) = x^32 * exp(16*x) * exp(8*x^2*exp(x)) * exp(4*F(2)) * exp(2*F(3)) * exp(F(4)),
...
F(n+1) = F(n)^2 * exp(F(n))
...
Then the e.g.f. A(x) equals the sum
A(x) = F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...
equivalently,
A(x) = x + x^2*exp(x) + x^4*exp(2*x)*exp(x^2*exp(x)) + x^8*exp(4*x)*exp(2*x^2*exp(x)) * exp(x^4*exp(2*x)*exp(x^2*exp(x))) + ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A369090, A369551 (a(n)/n), A030178.

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

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = x + A( x^2*exp(x) ).
(2) A(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) = G(x^2*exp(x))/x is the e.g.f. of A369090.

A073231 Decimal expansion of (1/e)^(1/e)^(1/e).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

			0.50047350056363684054513490133...

Cf. A001113 (e), A068985 (1/e), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073232 (((1/e)^(1/e))^(1/e)), A073227 (e^e^e).

With[{c=1/E},RealDigits[c^c^c,10,120][[1]]] (* Harvey P. Dale, Jul 16 2025 *)

PARI
```
exp(-1)^exp(-1)^exp(-1)
```

cp's OEIS Frontend

A277682 Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A277683 Decimal expansion of the modulus of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2*Pi*K*i.

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A126583 Decimal expansion of solution to exp(-x) = x^2.

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A198860 E.g.f. satisfies A(x) = 1/(1 - log(1 + x*A(x))).

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Maxima

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A245265 E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).

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A369090 Expansion of e.g.f. A(x) satisfying A(x) = A( x^2*exp(x) ) / x, with A(0) = 0.

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A299617 Decimal expansion of e^(W(1) + W(e)) = e/(W(1)*W(e)), where W is the Lambert W function (or PowerLog); see Comments.

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A277682 Decimal expansion of the imaginary part of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.

A277683 Decimal expansion of the modulus of the fixed point of exp(z) in C congruent with the branch K=1 of log(z)+2PiK*i.