A226571 -id:A226571 - cp's OEIS frontend

A006155 Expansion of e.g.f.: 1/(2-x-exp(x)).

1, 2, 9, 61, 551, 6221, 84285, 1332255, 24066691, 489100297, 11044268633, 274327080611, 7433424980943, 218208342366093, 6898241919264181, 233651576126946103, 8441657595745501019, 324052733365292875025, 13171257161208184782225, 565092918793429218839307

Offset: 0

Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..396
S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 157

Cf. A032112.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( 1/(2-x-Exp(x)) ))); // G. C. Greubel, Jan 09 2025

With[{nn=20},CoefficientList[Series[1/(2-x-E^x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 27 2018 *)

def A006155_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/(2-x-exp(x)) ).egf_to_ogf().list()
print(A006155_list(40)) # G. C. Greubel, Jan 09 2025

E.g.f.: 1/(2-x-exp(x)).
a(n) ~ n! / ((1+c) * (2-c)^(n+1)), where c = A226571 = LambertW(exp(2)). - Vaclav Kotesovec, Jun 06 2019
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020

More terms from Ralf Stephan, Mar 12 2004

A226572 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 12 2013

Let h(x) be the greater of the two solutions of s + log(s) = x; then this sequence represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = LambertW(exp(2)) = 1.5571455989976... is the unique root of the equation s + log(s) = 2. Equation s - log(s) = 2 does have two roots, but they are s = -LambertW(-1,-exp(-2)) = 3.14619322062... (this sequence) and s = -LambertW(-exp(-2)) = 0.158594339563... (A202348, not A226571). - Vaclav Kotesovec, Jan 09 2014]

Apart from the first digit the same as A202321. - R. J. Mathar, Jun 15 2013

			2 + log 2 = 2.693147...
2 + log(2 + log 2) = 2.990710...
2 + log(2 + log(2 + log 2)) = 3.095510...
limit(f(n)) = 3.14619322062...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A202321, A202348, A226571, A226573, A226574.

Cf. also A052526.

f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[2, 200]][[1]]  (* A226571 *)
d2 = RealDigits[g[2, 200]][[1]]  (* A226572 *)
s /. NSolve[s - Log[s] == 2, 200]  (* both constants *)
h[x_] := s /. NSolve[s - Log[s] == x]
Plot[h[x], {x, 1, 3}, PlotRange -> {0, 1}] (* bottom branch of h *)
Plot[h[x], {x, 1, 3}, PlotRange -> {1, 5}] (* top branch *)

default(realprecision, 100); solve(x=3, 4, x - log(x) - 2) \\ Jianing Song, Dec 30 2018

Equals -LambertW(-1, -exp(-2)) = A202321 + 2. - Vaclav Kotesovec, Jan 09 2014

Definition revised by N. J. A. Sloane, Dec 09 2017

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

A052842 E.g.f. A(x) = series reversion of (1-x)*(1-exp(-x)).

Original entry on oeis.org

0, 1, 3, 23, 290, 5104, 115374, 3185972, 103946688, 3912527016, 166884627360, 7955159511672, 419106982360560, 24182042474691984, 1516563901865906880, 102717031449780063360, 7472238163167018081024

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple grammar.

			E.g.f.: A(x) = x + 3*x^2/2! + 23*x^3/3! + 290*x^4/4! + 5104*x^5/5! +... which satisfies: A(x) = -log(1 - x/(1-A(x))).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 809
Vladimir Kruchinin, The method for obtaining expressions for coefficients of reverse generating functions, arXiv:1211.3244 [math.CO], 2012.

Cf. A226571.

spec := [S,{C=Prod(Z,B),S=Cycle(C),B=Sequence(S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
A052842 := proc (n) option remember; `if`(n = 0, 0, add(pochhammer(n, k)*abs(Stirling1(n, k+1)), k = 0..n-1)) end:
seq(A052842(n), n = 0..16); # Mélika Tebni, Jun 02 2023

CoefficientList[InverseSeries[Series[(-1 + E^(-x))*(x-1),{x,0,20}],x],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 08 2014 *)

a(n):=sum((sum((sum((stirling2(i+n-1,j)*binomial(j,j-i))/(i+n-1)!,i,0,j))*(-1)^(n+j-1)/(k-j)!,j,0,k))*(n+k-1)!,k,0,n-1); /* Vladimir Kruchinin, Feb 06 2012 */

{a(n)=n!*polcoeff(serreverse((1-exp(-x+O(x^(n+2))))*(1-x)),n)} /* Paul D. Hanna, Jun 22 2011 */

E.g.f. satisfies: A(x) = -log(1 - x/(1-A(x))). [From Encyclopedia of Combinatorial Structures]
a(n) = sum(k=0..n-1, (sum(j=0..k, (sum(i=0..j, (stirling2(i+n-1,j)*C(j,j-i))/ (i+n-1)!))*(-1)^(n+j-1)/(k-j)!))*(n+k-1)!), n>0. - Vladimir Kruchinin, Feb 06 2012
a(n) ~ n^(n-1) * c^n / (sqrt(1+c) * exp(n) * (c-1)^(2*n-1)), where c = LambertW(exp(2)) = 1.5571455989976114... (see A226571). - Vaclav Kotesovec, Jan 08 2014
For n >= 1, a(n) = Sum_{k=0..n-1} Pochhammer(n, k)*|Stirling1(n, k+1)|. - Mélika Tebni, Jun 02 2023

Name from a comment by Paul D. Hanna, Jun 22 2011

A226574 Decimal expansion of lim_{k->oo} f(k), where f(1)=e, and f(k) = e + log(f(k-1)) for k>1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 12 2013

Let g(x) be the greater of the two solutions of s + log(s) = x; then A226572 represents g(e). [See however the comments in A226571. - N. J. A. Sloane, Dec 09 2017]

			limit(f(n)) = 4.1386519474...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A226571, A226572, A226573.

f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[E, 200]][[1]]   (* A226573 *)
d2 = RealDigits[g[E, 200]][[1]]  (* A226574 *)

default(realprecision, 100); solve(x=4, 5, x - log(x) - exp(1)) \\ Jianing Song, Dec 24 2018

Equals -LambertW(-1, -exp(-e)). - Jianing Song, Dec 24 2018

Definition revised by N. J. A. Sloane, Dec 09 2017

A226573 Decimal expansion of lim_{k->oo} f(k), where f(1)=e, and f(k) = e - log(f(k-1)) for k>1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jun 12 2013

Old definition was: Decimal digits of limit(f(n)), where f(1) = e-log(e), f(n) = f(f(n-1)).

Let f(x) be lesser of the two solutions of s - log(s) = x; then A226571 represents f(e). [See however the comments in A226571. - N. J. A. Sloane, Dec 09 2017]

			limit(f(n)) = 2.0167797639...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A226571, A226572, A226574.

Program 1:
f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[E, 200]][[1]]  (* A226573 *)
d2 = RealDigits[g[E, 200]][[1]]  (* A226574 *)
s /. NSolve[s - Log[s] == 2, 200] (* both constants *)
***
Program 2:
N[ProductLog[E^E], 100] (* Clark Kimberling, Feb 15 2018 *)

default(realprecision, 100); lambertw(exp(exp(1))) \\ G. C. Greubel, Sep 09 2018

Equals LambertW(e^e). - Clark Kimberling, Feb 15 2018

Definition revised by N. J. A. Sloane, Dec 09 2017

A032182 "CIJ" (necklace, indistinct, labeled) transform of 2,1,1,1...

Original entry on oeis.org

2, 5, 23, 156, 1409, 15908, 215529, 3406770, 61542033, 1250700672, 28241802929, 701494286870, 19008349975497, 557990502346020, 17639808964667241, 597481099591875594, 21586547569205365409, 828650020968205107752, 33680822283049282596225

Offset: 1

Christian G. Bower

nonn

Christian G. Bower, Transforms (2)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 837
Index entries for sequences related to necklaces

Cf. A226571.

E.g.f.: A(x) = log(1/(2-x-exp(x))). - Sean A. Irvine, May 28 2020

a(n) ~ (n-1)! / (2 - LambertW(exp(2)))^n. - Vaclav Kotesovec, May 29 2020

More terms from Sean A. Irvine, May 28 2020

A207493 E.g.f. A(x) is the series reversion of 2x-1/2x^2-exp(x)+1.

Original entry on oeis.org

1, 2, 13, 141, 2141, 41798, 997340, 28124253, 915095222, 33744966795, 1390772973547, 63353273661835, 3160751396077900, 171405094563763674, 10038777321831260503, 631498191927510881178, 42464602911622645539047, 3039724643022777390236243

Offset: 1

Vladimir Kruchinin, Feb 18 2012

nonn

Cf. A226571.

Rest[CoefficientList[InverseSeries[Series[2*x-1/2*x^2-E^x+1, {x, 0, 20}], x],x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 22 2014 *)

a(n):=(sum((n+k-1)!*sum(1/(k-j)!*sum(1/l!*sum(((-1)^(i+l)*2^(l-2*i) *binomial(l,i)*stirling2(n+j-i-l-1,j-l))/(n+j-i-l-1)!, i,0,l), l,0,j), j,0,k), k,0,n-1));

a(n) = (sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, 1/l!*sum(i=0..l, ((-1)^(i+l)*2^(l-2*i)* C(l,i)*stirling2(n+j-i-l-1,j-l))/(n+j-i-l-1)!))))).

a(n) ~ n^(n-1) / (sqrt(1+c) * exp(n) * (3-c*(2+c)/2)^(n-1/2)), where c = LambertW(exp(2)) = 1.5571455989976... (see A226571). - Vaclav Kotesovec, Jan 22 2014

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A006155 Expansion of e.g.f.: 1/(2-x-exp(x)).

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A226572 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 + log(f(k-1)) for k>1.

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

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A052842 E.g.f. A(x) = series reversion of (1-x)*(1-exp(-x)).

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A226574 Decimal expansion of lim_{k->oo} f(k), where f(1)=e, and f(k) = e + log(f(k-1)) for k>1.

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A226573 Decimal expansion of lim_{k->oo} f(k), where f(1)=e, and f(k) = e - log(f(k-1)) for k>1.

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A207493 E.g.f. A(x) is the series reversion of 2x-1/2x^2-exp(x)+1.