A052813 -id:A052813 - cp's OEIS frontend

A052807 Expansion of e.g.f. -LambertW(log(1-x)).

0, 1, 3, 17, 146, 1704, 25284, 456224, 9702776, 237711888, 6593032560, 204212077992, 6986942528400, 261700394006232, 10650713784774504, 468007296229553880, 22083086552247101184, 1113646609708909274880

Offset: 0

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

Previous name was: A simple grammar.

E.g.f. of A052813 equals exp(A(x)) = -A(x)/log(1-x). a(n) = n!*Sum_{k=0..n-1} A052813(k)/k!/(n-k). - Paul D. Hanna, Jul 19 2006

			E.g.f.: A(x) = x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! +...
A(x)/exp(A(x)) = -log(1-x) = x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 +...

G. C. Greubel, Table of n, a(n) for n = 0..375
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 769

Cf. A006963, A048802, A052813 (exp(A(x))), A277489.

spec := [S,{B=Cycle(Z),C=Set(S),S=Prod(C,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

max = 17; se = Series[-ProductLog[-Log[-(-1 + x)^(-1)]] , {x, 0, max}]; Join[{0}, (CoefficientList[se, x] // DeleteCases[#, 0] &) * Range[max]!] (* Jean-François Alcover, Jun 24 2013 *)
CoefficientList[Series[-LambertW[-Log[-1/(-1 + x)]], {x,0,50}], x]*
Range[0,50]! (* G. C. Greubel, Jun 18 2017 *)

{a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))^A);n!*polcoeff(log(A),n)} \\ Paul D. Hanna, Jul 19 2006

x = 'x + O('x^30); concat(0, Vec(serlaplace(-lambertw(log(1-x))))) \\ Michel Marcus, Jun 19 2017

a(n) = Sum_{k=1..n} |Stirling1(n, k)|*k^(k-1). - Vladeta Jovovic, Sep 17 2003
E.g.f. satisfies: A(x) = 1/(1-x)^A(x). - Paul D. Hanna, Jul 19 2006
a(n) ~ n^(n-1)*exp((exp(-1)-1)*n+1/2) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013
E.g.f.: Series_Reversion( 1 - exp(-x*exp(-x)) ). - Seiichi Manyama, Sep 08 2024

New name using e.g.f. by Vaclav Kotesovec, Oct 18 2013

A349556 E.g.f. satisfies: A(x) = 1/(1 - x*A(x))^A(x).

Original entry on oeis.org

1, 1, 6, 69, 1196, 27900, 820554, 29168048, 1216826120, 58301363808, 3155539049040, 190434409300872, 12679792851087768, 923409652630222680, 73016802381788896344, 6230201355664856039640, 570574779781503603910464, 55826084651771645745562368

Offset: 0

Seiichi Manyama, Nov 21 2021

nonn

Cf. A052813, A162655, A349559.

a[n_] := Sum[(n + k + 1)^(k - 1) * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 22 2021 *)

a(n) = sum(k=0, n, (n+k+1)^(k-1)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (n+k+1)^(k-1) * |Stirling1(n,k)|.

a(n) ~ s^2 * sqrt((1 - r*s) / (1 + r*s*(s-1) * (2 - r*s))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.1591040778917510493879632960549533860431737829556... and s = 1.588466710327904339474066925768589168215650366378... are real roots of the system of equations 1/s = (1 - r*s)^s, r*s/(1 - r*s) - log(1 - r*s) = 1/s. - Vaclav Kotesovec, Nov 22 2021

A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

Original entry on oeis.org

1, 1, -2, 9, -68, 740, -10554, 185906, -3891320, 94259952, -2592071760, 79748398752, -2713685928744, 101184283477680, -4102325527316184, 179674073609647080, -8454031849605513024, 425281651659459346944, -22777115050468598701248

Offset: 0

Paul D. Hanna, Jul 20 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A008275, A052813, A052807, A349561, A349583, A349585.

CoefficientList[Series[Log[1+x]/LambertW[Log[1+x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Table[StirlingS1[n, 0] + StirlingS1[n, 1] + Sum[(-1)^(k + 1)*StirlingS1[n, k]*(k - 1)^(k - 1), {k, 2, n}], {n,0,50}] (* G. C. Greubel, Jun 21 2017 *)
CoefficientList[Series[Exp[LambertW[Log[1+x]]], {x, 0, 25}], x]* Range[0, 25]! (* G. C. Greubel, Jun 22 2017 *)

{a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0); A[ #A]=-Vec(Ser(A)^Ser(A))[ #A]);n!*A[n+1]}

x='x+O('x^50); Vec(serlaplace(exp(lambertw(log(1+x))))) \\ G. C. Greubel, Jun 22 2017

E.g.f.: A(x) = log(1+x)/LambertW(log(1+x)).
log(A(x)) = LambertW(log(1+x)).
E.g.f.: A(x) = 1/G(-x) where G(x) = g.f. of A052813.
E.g.f. of A052807 = -log(A(-x)) = -log(1-x)/A(-x).
a(n) = Sum_{k=0..n} (-1)^(k+1)*Stirling1(n,k)*(k-1)^(k-1). - Vladeta Jovovic, Jul 22 2006
|a(n)| ~ exp((exp(-1)-1)*n+3/2) * n^(n-1) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

A355842 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

Original entry on oeis.org

1, 0, 2, 3, 44, 210, 3054, 27300, 449952, 6020784, 115381080, 2053568880, 45733246536, 1010390340960, 25916586868704, 680621684914080, 19881379012231680, 603034125051738240, 19833651290982164544, 680927283288289169280, 24953207662252739030400

Offset: 0

Seiichi Manyama, Jul 18 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A052813, A184949, A355779.

nmax = 20; CoefficientList[Series[LambertW[x * Log[1-x]] / (x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 21 2022 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x*log(1-x)))))

a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, Aug 28 2022

E.g.f.: exp( -LambertW(x * log(1-x)) ).
E.g.f.: LambertW(x * log(1-x)) / (x * log(1-x)).
a(n) ~ sqrt(1 + exp(1)*r^2/(1-r)) * n^(n-1) / (exp(n-1) * r^n), where r = 0.5123112855238643734867005914814802444318611742265... is the positive root of the equation r*log(1-r) = -exp(-1). - Vaclav Kotesovec, Jul 21 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 28 2022

A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).

Original entry on oeis.org

1, 1, 8, 123, 2880, 91380, 3670422, 178637046, 10220188584, 672297233904, 49998246211440, 4148400045354048, 379922492043352728, 38069325641907527760, 4143052833221549219112, 486645149764644576491160, 61364015736691449905270208, 8267876533375590035644494336, 1185420892076469924434688605760

Offset: 0

Paul D. Hanna, Nov 18 2015

nonn

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 123*x^3/3! + 2880*x^4/4! + 91380*x^5/5! + 3670422*x^6/6! + 178637046*x^7/7! + 10220188584*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^3).
From a LambertW identity,
A(x) = 1 - log(1-x) + 7*log(1-x)^2/2! - 10^2*log(1-x)^3/3! + 13^3*log(1-x)^4/4! - 16^4*log(1-x)^5/5! + 19^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^3 + x^2*A(x)^3*(A(x)^3+1)/2! + x^3*A(x)^3*(A(x)^3+1)*(A(x)^3+2)/3! + x^4*A(x)^3*(A(x)^3+1)*(A(x)^3+2)*(A(x)^3+3)/4! +...

Seiichi Manyama, Table of n, a(n) for n = 0..331

Cf. A052813, A264407, A349505.

Table[Sum[Abs[StirlingS1[n, k]] * (3*k+1)^(k-1), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 18 2015 *)

{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^3 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = sum(k=0, n, abs(stirling(n, k, 1))*(3*k+1)^(k-1));
for(n=0, 20, print1(a(n), ", "))

E.g.f. also satisfies:
(1) A(x) = ( LambertW(3*log(1-x)) / (3*log(1-x)) )^(1/3).
(2) A(x) = Sum_{n>=0} (3*n+1)^(n-1) * (-log(1-x))^n / n!.
(3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^3 + k).
(4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(3*k).
a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (3*k+1)^(k-1).
a(n) ~ n^(n-1) / (sqrt(3) * exp(n - n*exp(-1)/3 - 5/6) * (exp(exp(-1)/3) - 1)^(n-1/2)). - Vaclav Kotesovec, Nov 18 2015

A264407 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^2).

Original entry on oeis.org

1, 1, 6, 66, 1084, 23920, 665388, 22374884, 883177328, 40043323728, 2051202965280, 117166763184768, 7384596609153696, 509084508866799840, 38108295339435463296, 3078340850588419228800, 266906341797637061659392, 24724454378396015985551616, 2436960508983873399401081856, 254658073346711773211982974976, 28122779871625104764662272952320

Offset: 0

Paul D. Hanna, Nov 17 2015

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1084*x^4/4! + 23920*x^5/5! + 665388*x^6/6! + 22374884*x^7/7! + 883177328*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^2).
From a LambertW identity,
A(x) = 1 - log(1-x) + 5*log(1-x)^2/2! - 7^2*log(1-x)^3/3! + 9^3*log(1-x)^4/4! - 11^4*log(1-x)^5/5! + 13^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^2 + x^2*A(x)^2*(A(x)^2+1)/2! + x^3*A(x)^2*(A(x)^2+1)*(A(x)^2+2)/3! + x^4*A(x)^2*(A(x)^2+1)*(A(x)^2+2)*(A(x)^2+3)/4! +...

Seiichi Manyama, Table of n, a(n) for n = 0..347

Cf. A052813, A264408, A349504.

Table[Sum[Abs[StirlingS1[n, k]] * (2*k+1)^(k-1), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 18 2015 *)

{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^2 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = sum(k=0,n,abs(stirling(n, k, 1))*(2*k+1)^(k-1));
for(n=0,20,print1(a(n),", "))

E.g.f. also satisfies:
(1) A(x) = sqrt( LambertW(2*log(1-x)) / (2*log(1-x)) ).
(2) A(x) = Sum_{n>=0} (2*n+1)^(n-1) * (-log(1-x))^n / n!.
(3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^2 + k).
(4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(2*k).
a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (2*k+1)^(k-1).
a(n) ~ exp(1 + (exp(-1)/2 - 1)*n) * n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n-1/2)). - Vaclav Kotesovec, Nov 18 2015

A356960 E.g.f. satisfies: A(x) = 1/(1 - x * A(x)^3)^A(x).

Original entry on oeis.org

1, 1, 10, 207, 6620, 288040, 15891234, 1063219640, 83665143176, 7572321823536, 775010639465040, 88510236140283672, 11158965455394331992, 1539441941412714237912, 230675631266761375815288, 37309025609545822539225240, 6478248637390494598048444224

Offset: 0

Seiichi Manyama, Sep 07 2022

nonn

Sidney Cadot, Table of n, a(n) for n = 0..100

Cf. A052813, A349556, A356914.

Cf. A356973.

a[n_] := Sum[(3*n + k + 1)^(k - 1)*Abs[StirlingS1[n, k]], {k, 0, n}] (* Sidney Cadot, Jan 05 2023 *)

a(n) = sum(k=0, n, (3*n+k+1)^(k-1)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} (3*n+k+1)^(k-1) * |Stirling1(n,k)|.

A355779 E.g.f. satisfies A(x) = 1/(1 - x)^(2 * A(x)).

Original entry on oeis.org

1, 2, 14, 168, 2912, 66600, 1900056, 65101120, 2606993728, 119561789952, 6181730106240, 355838533286016, 22573258090527360, 1564818434983235328, 117698836976753297664, 9547346757806586746880, 830846347686871026714624, 77215374643802544102187008

Offset: 0

Seiichi Manyama, Jul 16 2022

nonn

Cf. A052813, A351275, A355786.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(2*log(1-x)))))

a(n) = sum(k=0, n, 2^k*(k+1)^(k-1)*abs(stirling(n, k, 1)));

E.g.f.: exp( -LambertW(2 * log(1-x)) ).
a(n) = Sum_{k=0..n} 2^k * (k+1)^(k-1) * |Stirling1(n,k)|.
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: LambertW(2 * log(1-x)) / (2 * log(1-x)).
a(n) ~ sqrt(2) * n^(n-1) * exp(3/2 - n + n*exp(-1)/2) / (exp(exp(-1)/2) - 1)^(n - 1/2). (End)

A362796 E.g.f. satisfies A(x) = 1/(1-x)^(A(x)^x).

Original entry on oeis.org

1, 1, 2, 12, 72, 650, 6480, 80906, 1121512, 18069264, 320204160, 6348340152, 136915211664, 3230148306216, 82078412377416, 2248247450065080, 65771634671679360, 2052879248516927232, 67955959831214467584, 2381716543764159438336

Offset: 0

Seiichi Manyama, May 04 2023

nonn

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

Cf. A052813, A362798.

Cf. A362794, A362799.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^exp(-lambertw(x*log(1-x)))))

E.g.f.: exp( -LambertW(x * log(1-x)) / x ) = 1/(1-x)^exp( -LambertW(x * log(1-x)) ).

E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (-log(1-x))^k / k!.

A362798 E.g.f. satisfies A(x) = 1/(1-x)^(A(x)^(x^2)).

Original entry on oeis.org

1, 1, 2, 6, 48, 360, 2820, 31500, 393568, 5111568, 78491520, 1345893120, 24286008384, 483716087712, 10526811186528, 241867328844960, 5957816820215040, 157412355684364800, 4380674530640290560, 128826276098289179904, 4010282529115722232320

Offset: 0

Seiichi Manyama, May 04 2023

nonn

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

Cf. A052813, A362796.

Cf. A362795, A362800.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^exp(-lambertw(x^2*log(1-x)))))

E.g.f.: exp( -LambertW(x^2 * log(1-x)) / x^2 ) = 1/(1-x)^exp( -LambertW(x^2 * log(1-x)) ).

E.g.f.: Sum_{k>=0} (k*x^2 + 1)^(k-1) * (-log(1-x))^k / k!.

cp's OEIS Frontend

A052807 Expansion of e.g.f. -LambertW(log(1-x)).

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A349556 E.g.f. satisfies: A(x) = 1/(1 - x*A(x))^A(x).

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A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

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A355842 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

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A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).

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A264407 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^2).

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A356960 E.g.f. satisfies: A(x) = 1/(1 - x * A(x)^3)^A(x).

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