A202356 -id:A202356 - cp's OEIS frontend

A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

1, 1, 3, 19, 181, 2321, 37501, 731935, 16758393, 440525377, 13077834841, 432796650551, 15799794395749, 630773263606513, 27339525297079269, 1278550150117141231, 64171287394646697841, 3440711053857464325377

Offset: 0

Paul D. Hanna, Jun 17 2009

nonn

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...
log(A(x))/x = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)^3 + x^4*A(x)^4 +...

G. C. Greubel, Table of n, a(n) for n = 0..373
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Cf. A161633 (e.g.f. = log(A(x))/x).

Cf. A212722, A212917, A245265, A125500.

Table[Sum[n! * (n-k+1)^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n,m=1)=if(n==0,1,sum(k=0,n,n!/k!*m*(n-k+m)^(k-1)*binomial(n-1,n-k)))}

{a(n,m=1)=local(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)));n!*polcoeff(A^m,n)}

a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(n-k+m)^(k-1)/k! * C(n-1,n-k).
E.g.f. satisfies: A(x) = exp(x) * A(x)^(x*A(x)). - Paul D. Hanna, Aug 02 2013
a(n) ~ n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2) * exp(n) * c^(2*n+3/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A202322 Decimal expansion of x satisfying x+2=exp(-x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 18 2011

For many choices of u and v, there is just one value of x satisfying u*x+v=e^(-x).  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 2.... A202322
1.... 3.... A202323
2.... 2.... A202353
2.... e.... A202354
1... -1.... A202355
1.... 0.... A030178
2.... 0.... A202356
e.... 0.... A202357
3.... 0.... A202392
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0.  We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A202322, take f(x,u,v)=x+2-e^(-x) and g(u,v) = a nonzero solution x of f(x,u,v)=0.  If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous.  A portion of an implicit surface is plotted by Program 2 in the Mathematica section.

			x=-0.442854401002388583141327999999336819716262...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for transcendental numbers

Cf. A202320.

(* Program 1:  A202322 *)
u = 1; v = 2;
f[x_] := u*x + v; g[x_] := E^-x
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.45, -.44}, WorkingPrecision -> 110]
RealDigits[r]  (* A202322 *)
(* Program 2: implicit surface of u*x+v=e^(-x) *)
f[{x_, u_, v_}] := u*x + v - E^-x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 2}]}, {v, 1, 3}, {u, 1, 3}];
ListPlot3D[Flatten[t, 1]] (* for A202322 *)
RealDigits[ ProductLog[E^2] - 2, 10, 99] // First (* Jean-François Alcover, Feb 14 2013 *)

lambertw(exp(2)) - 2 \\ G. C. Greubel, Jun 10 2017

x(u,v) = W(e^(v/u)/u) - v/u, where W = ProductLog = LambertW. - Jean-François Alcover, Feb 14 2013

Equals A226571 - 2 = LambertW(exp(2))-2. - Vaclav Kotesovec, Jan 09 2014

Digits from a(84) on corrected by Jean-François Alcover, Feb 14 2013

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

A200319 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2).

Original entry on oeis.org

1, 2, 12, 132, 2040, 40440, 979440, 28034160, 925858080, 34654465440, 1449705660480, 67029745527360, 3394417068282240, 186842736763562880, 11107390768144070400, 709223357051739129600, 48408150749346010022400, 3517279496138031162739200, 271050342684747077612160000

Offset: 1

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 132*x^4/4! + 2040*x^5/5! +...
where A(1+x - exp(x^2)) = x and A(x) = x-1 + exp(A(x)^2).

Cf. A200320, A213640, A202356.

Rest[CoefficientList[InverseSeries[Series[1 - E^x^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+x^2*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, (exp(x^2+x*O(x^n))-1)^m)/m!); n!*polcoeff(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, (exp(x^2+x*O(x^n))-1)^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}

E.g.f.: Series_Reversion(1+x - exp(x^2)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x^2)-1)^n/x / n! ).
a(n) ~ (c/2)^(1/4) * n^(n-1) / (sqrt(1+c) * exp(n) * (1+sqrt(c/2)-1/sqrt(2*c))^(n-1/2)), where c = LambertW(1/2) = 0.351733711249195826... (see A202356). - Vaclav Kotesovec, Jan 10 2014

A099554 Decimal expansion of the constant x that satisfies x = exp(1/sqrt(x)).

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Oct 22 2004

This constant arises from the series: S(n) = Sum_{k=0..2n} (n-[k/2])^k/k!. The asymptotic behavior of this series is given by: S(n) ~ c*x^n where c = (x+sqrt(x))/(1+2*sqrt(x)) = 0.8957126... and x = 2.0207473586... satisfies x = exp(1/sqrt(x)).

			x=2.02074735861185766811269528724732366499433113141625298973171160826928577...
To demonstrate how this constant describes the asymptotics of the sum:
S(n) = Sum_{k=0..2n} (n-[k/2])^k/k! ~ c*x^n
evaluate the sum at n=5:
S(5) = 1+ 5+ 4^2/2!+ 4^3/3!+ 3^4/4!+ 3^5/5!+ 2^6/6!+ 2^7/7!+ 1/8!+ 1/9!
= 782291/25920 = 30.1809799... = (0.89572199...)*x^5
and evaluate the sum at n=6:
S(6) = 1+ 6+ 5^2/2!+ 5^3/3!+ 4^4/4!+ 4^5/5!+ 3^6/6!+ 3^7/7!+ 2^8/8!+ 2^9/9!+ 1/10!+ 1/11!
= 608606683/9979200 = 60.9875223... = (0.89571298...)*x^6.

Cf. A099555, A202356.

RealDigits[x/.FindRoot[x==Exp[1/Sqrt[x]],{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jan 06 2013 *)
RealDigits[ 1/(4*ProductLog[1/2]^2), 10, 105] // First (* Jean-François Alcover, Feb 15 2013 *)

PARI
```
solve(x=2,2.1,x-exp(1/sqrt(x)))
```

Equals 1/(4*A202356^2). - Vaclav Kotesovec, Oct 06 2020

A143154 E.g.f.: A(x) = x + log(1 - A(x))^2.

Original entry on oeis.org

1, 2, 18, 262, 5320, 138728, 4419156, 166319424, 7221397848, 355312006392, 19537581248592, 1187337791554176, 79025863405440432, 5716937001401316000, 446654003380859659488, 37480492611898380241248

Offset: 1

Paul D. Hanna, Jul 27 2008

nonn

Radius of convergence is r = (-1 + 6*A(r) - A(r)^2)/4 = 0.172815973872...

where A(r) = 1 - exp((A(r)-1)/2) = 0.2965325775...

			A(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! + ...
-log(1 - A(x)) = G(x) = the g.f. of A143155:
G(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! + ...
G(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! + ...
Related expansions:
A(x) = x + log(1-x)^2 + d/dx log(1-x)^4/2! + d^2/dx^2 log(1-x)^6/3! + d^3/dx^3 log(1-x)^8/4! + ...
log(A(x)/x) = log(1-x)^2/x + d/dx (log(1-x)^4/x)/2! + d^2/dx^2 (log(1-x)^6/x)/3! + d^3/dx^3 (log(1-x)^8/x)/4! + ...

Cf. A143155, A143138, A202356.

Rest[CoefficientList[InverseSeries[Series[x-Log[1-x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 24 2014 *)

a(n):=((n-1)!*sum(binomial(n+k-1,n-1)*sum((-1)^(n+j-1)*binomial(k,j)*sum((binomial(j,l)*(2*(j-l))!*stirling1(n-l+j-1,2*(j-l)))/(n-l+j-1)!,l,0,j),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 07 2012 */

{a(n)=local(A=x+O(x^n));for(i=0,n,A=x + log(1-A)^2);n!*polcoeff(A,n)}

{a(n)=n!*polcoeff(serreverse(x-log(1-x+x*O(x^n))^2),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, log(1-x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(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, log(1-x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

E.g.f.: A(x) = Series_Reversion( x - log(1 - x)^2 ).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) log(1-x)^(2*n)/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (log(1-x)^(2*n)/x)/n! ).
E.g.f. derivative: A'(x) = (1 - A(x))/(1 - A(x) + 2*log(1 - A(x))).
a(n) = ((n-1)!*sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(n+j-1)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*stirling1(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. - Vladimir Kruchinin, Feb 07 2012
a(n) ~ c*sqrt(4/(1+c)-2-2*c) * n^(n-1) / (exp(n) * (1-c*(2+c))^n), where c = LambertW(1/2) = 0.35173371124919... (see A202356). - Vaclav Kotesovec, Jan 24 2014

A143155 E.g.f.: A(x) = -log(1 - x - A(x)^2).

Original entry on oeis.org

1, 3, 26, 376, 7614, 198248, 6309092, 237291388, 10297903920, 506495785632, 27842563031304, 1691646018671376, 112569103111005072, 8142200129607522288, 636046143210331062048, 53366672768969064921024

Offset: 1

Paul D. Hanna, Jul 27 2008

nonn

			A(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! +...
x + A(x)^2 = 1 - exp(-A(x)) = G(x) = g.f. of A143154:
G(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! +...

Cf. A143139, A143154, A202356.

Table[Sum[(n+k-1)!*Sum[(-1)^(n+j-1)/(k-j)!*Sum[(StirlingS2[n-2*l+j-1,j-l])/(l!*(n-2*l+j-1)!),{l,0,Min[j,(n+j-1)/2]}],{j,0,k}],{k,0,n-1}],{n,1,20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Dec 28 2013 *)

a(n):=(sum((n+k-1)!*sum((-1)^(n+j-1)/(k-j)!*sum((stirling2(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!),l,0,min(j,(n+j-1)/2)),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 03 2012 */

{a(n)=local(A=x+O(x^n));for(i=0,n,A=-log(1-x-A^2));n!*polcoeff(A,n)}

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

a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} ((-1)^(n+j-1)/(k-j)!)*Sum_{L=0..min(j, (n+j-1)/2)} Stirling2(n-2*L+j-1, j-l)/(L!*(n-2*L+j-1)!), n > 0. - Vladimir Kruchinin, Feb 03 2012

a(n) ~ n^(n-1) / (sqrt(2*(1+c)) * exp(n) * (1-2*c-c^2)^(n-1/2)), where c = LambertW(1/2). - Vaclav Kotesovec, Dec 28 2013

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A161630 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)) ).

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A202322 Decimal expansion of x satisfying x+2=exp(-x).

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

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A200319 E.g.f. satisfies: A(x) = x-1 + exp(A(x)^2).

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A099554 Decimal expansion of the constant x that satisfies x = exp(1/sqrt(x)).

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A143154 E.g.f.: A(x) = x + log(1 - A(x))^2.

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A143155 E.g.f.: A(x) = -log(1 - x - A(x)^2).

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