A349561 -id:A349561 - cp's OEIS frontend

A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

1, 1, 0, 2, -9, 72, -710, 8563, -121814, 1997502, -37097739, 769687954, -17644355410, 442894514285, -12081668234012, 355889274553166, -11258683640579857, 380701046875217492, -13702507978018209458, 523049811008797507683, -21105565578064063658754

Offset: 0

Seiichi Manyama, Nov 22 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..399
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A120980, A349561, A349585.

Cf. A008277, A052880, A349524, A349525.

b:= proc(n, m) option remember; `if`(n=0,
     (1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

a(n) = Sum_{k=0..n} (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(exp(x) - 1) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) ~ -(-1)^n * sqrt(exp(1) - 1) * n^(n-1) / (exp(n+1) * (1 - log(exp(1) - 1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 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

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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

A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

Original entry on oeis.org

1, 1, -2, 8, -59, 642, -9112, 158839, -3279880, 78250188, -2117569181, 64082989720, -2144319848772, 78609355884893, -3133061858717806, 134884905211588892, -6238095343894356675, 308427209934965151158, -16234730389499986865092, 906409067599064528054343

Offset: 0

Seiichi Manyama, Nov 22 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..378
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A120980, A349561, A349583.

Cf. A008277, A058864, A349527, A349528.

b:= proc(n, m) option remember; `if`(n=0,
     (m-1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> (-1)^(n-1)*b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 03 2022

a[n_] := (-1)^(n - 1) * Sum[If[k == 1, 1, (k - 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(1 - exp(-x)) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ -(-1)^n * sqrt(1 + exp(1)) * n^(n-1) / (exp(n+1) * (log(1 + exp(1)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A349652 E.g.f. satisfies: A(x)^(A(x)^2) = 1/(1 - x).

Original entry on oeis.org

1, 1, -2, 18, -220, 3880, -86868, 2376836, -76735216, 2856604464, -120457684320, 5675047644288, -295430737430112, 16840861797433440, -1043322313406139648, 69798144929293516800, -5014888682767294232832, 385130588783629323238656

Offset: 0

Seiichi Manyama, Nov 23 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..356
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349650, A349654, A349656.

Cf. A349561, A349653.

nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1 - x))^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (2*k-1)^(k-1)*log(1-x)^k/k!)))

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

a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = -Sum_{k>=0} (2*k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = ( -2*log(1-x)/LambertW(-2*log(1-x)) )^(1/2).
a(n) ~ -(-1)^n * n^(n-1) / (sqrt(2) * exp(n + exp(-1)/4) * (-1 + exp(exp(-1)/2))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021

A229237 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^4).

Original entry on oeis.org

1, 1, 8, 135, 3544, 126980, 5778606, 319234454, 20755549256, 1552791269232, 131408062049040, 12411898074678432, 1294418444771718168, 147733436055601473168, 18315901821846419101416, 2451257290708213030681080, 352217918432527724627871936, 54082428426583359310449351168

Offset: 0

Vaclav Kotesovec, Sep 17 2013

nonn

Generally, for A(x)^A(x) = 1/(1-x*A(x)^p) is limit n->infinity a(n)^(1/n)/n = exp(p*(1-r)/(r-p))*(p-r+exp(r/(p-r))), where r is the root of the equation exp(r/(p-r)) = (r-p)/r*(r + LambertW(-1,-r*exp(-r))

Generally, if e.g.f. A(x) satisfies A(x)^A(x) = 1/(1-x*A(x)^p), then a(n) ~ s*sqrt((s^s-1)/(p*(s^s-1)*(p*s^s-1)-s)) * n^(n-1) * (s^(p+s)/(s^s-1))^n / exp(n), where s is the root of the equation (1+log(s))*s = (s^s-1)*p. Compared with my previous result, limit n->infinity a(n)^(1/n)/n = s^(p+s)/(s^s-1)/exp(1). - Vaclav Kotesovec, Dec 28 2013

Seiichi Manyama, Table of n, a(n) for n = 0..326

Cf. A141209, A216135, A216136, A349561.

Table[Sum[(4*n-k+1)^(k-1)*(-1)^(n-k)*StirlingS1[n,k],{k,0,n}],{n,0,20}]
p=4; E^(p*(1-r)/(r-p))*(p-r+E^(r/(p-r)))/.FindRoot[E^(r/(p-r))==(r-p)/r*(r+LambertW[-1,-r*E^(-r)]), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n *)

Limit n->infinity a(n)^(1/n)/n = exp(4*(1-r)/(r-4))*(4-r+exp(r/(4-r))) = 3.635561077783029..., where r = 0.8373821681637... is the root of the equation exp(r/(4-r)) = (r-4)/r*(r + LambertW(-1,-r*exp(-r))

a(n) ~ s*sqrt((s^s-1)/(4*(s^s-1)*(4*s^s-1)-s)) * n^(n-1) * (s^(4+s)/(s^s-1))^n / exp(n), where s = 1.3031377498774256189193761312... is the root of the equation (1+log(s))*s = 4*(s^s-1). - Vaclav Kotesovec, Dec 28 2013

A349653 E.g.f. satisfies: A(x)^(A(x)^3) = 1/(1 - x).

Original entry on oeis.org

1, 1, -4, 51, -996, 27120, -943602, 40023354, -2002953432, 115536775248, -7547711366880, 550798542893808, -44409102801760584, 3920444594317227600, -376109365694009875704, 38961901445878423746360, -4334496557343337848950208, 515407133679990302374396416

Offset: 0

Seiichi Manyama, Nov 23 2021

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Seiichi Manyama, Table of n, a(n) for n = 0..336
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349651, A349655, A349657.

Cf. A349561, A349652.

nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1 - x))^(1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*log(1-x)^k/k!)))

my(N=20, x='x+O('x^N)); Vec(serlaplace((-3*log(1-x)/lambertw(-3*log(1-x)))^(1/3)))

a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * Stirling1(n,k).
E.g.f.: A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = ( -3*log(1-x)/LambertW(-3*log(1-x)) )^(1/3).
a(n) ~ -(-1)^n * exp(1/6 - exp(-1)/6 - n) * n^(n-1) / (sqrt(3) * (-1 + exp(exp(-1)/3))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021

A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.

Original entry on oeis.org

1, 0, 2, 3, -4, -30, 294, 3780, -7904, -444528, 78840, 99657360, 539299848, -27852945120, -361237078944, 10124338180320, 258341121976320, -4020500134465920, -205187357182405824, 1330097523844832640, 186823640933648588160, 500469438126120583680

Offset: 0

Seiichi Manyama, Sep 03 2022

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

Cf. A355842, A356795, A356796, A356906.

Cf. A349561, A356884.

nmax = 21; A[_] = 1;
Do[A[x_] = (1/(1 - x)^x)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x*log(1-x))^k/k!)))

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

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

a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x * log(1-x)) ).
E.g.f.: A(x) = -x * log(1-x)/LambertW(-x * log(1-x)).

A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.

Original entry on oeis.org

1, 2, -2, 24, -272, 4840, -107496, 2934400, -94501760, 3511914624, -147882696960, 6959217277056, -361941871760256, 20616017227643136, -1276341094954066176, 85337532623368181760, -6128269375791673718784, 470426144496265208979456

Offset: 0

Seiichi Manyama, Sep 03 2022

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Robert Israel, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349561, A355779, A356909.

S:= series(exp( LambertW(-2 * log(1-x))),x,51):
seq(n!*coeff(S,x,n),n=0..50); # Robert Israel, Sep 18 2022

nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1-x)^2)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-2*log(1-x))^k/k!)))

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

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

a(n) = Sum_{k=0..n} 2^k * (-k+1)^(k-1) * |Stirling1(n,k)|.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-2 * log(1-x)) ).
E.g.f.: A(x) = -2 * log(1-x)/LambertW(-2 * log(1-x)).

A357243 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).

Original entry on oeis.org

1, 1, -2, 6, -52, 540, -7608, 129304, -2612608, 60867360, -1608663840, 47527158624, -1552431588288, 55547889458880, -2160724031160576, 90782738645280000, -4097139872604807168, 197675862365363088384, -10153243488783257091072

Offset: 0

Seiichi Manyama, Sep 19 2022

sign

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

Cf. A005727, A155456, A349561, A356905, A356908.

nmax = 20; A[_] = 1;
Do[A[x_] = ((1 - x)^(-1 + x))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-(1-x)*log(1-x))^k/k!)))

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

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

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

cp's OEIS Frontend

A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

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

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A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

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A349652 E.g.f. satisfies: A(x)^(A(x)^2) = 1/(1 - x).

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A229237 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^4).

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A349653 E.g.f. satisfies: A(x)^(A(x)^3) = 1/(1 - x).

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A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.

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