A376100 -id:A376100 - cp's OEIS frontend

A376106 Expansion of e.g.f. LambertW(x / (1 - 2*x)).

0, 1, 2, 9, 56, 465, 4764, 58345, 830192, 13466817, 245254580, 4955259441, 109995693576, 2661003245329, 69682488950060, 1963774182830265, 59261538449833184, 1906643335934717697, 65149411890671521380, 2356212733788818122561, 89920484394446094721400

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A060356, A376107.

Cf. A376100.

nmax=20; CoefficientList[InverseSeries[Series[x / (2*x + E^(-x)), {x, 0, nmax}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 20 2024 *)

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(lambertw(x/(1-2*x)))))

a(n) = n!*sum(k=1, n, 2^(n-k)*(-k)^(k-1)*binomial(n-1, k-1)/k!);

E.g.f. A(x) satisfies A(x) = x * (2*A(x) + exp(-A(x))).
E.g.f.: Series_Reversion( x / (2*x + exp(-x)) ).
a(n) = n! * Sum_{k=1..n} 2^(n-k) * (-k)^(k-1) * binomial(n-1,k-1)/k!.

A376101 Expansion of e.g.f. -LambertW(-x / (1 - 3*x)).

Original entry on oeis.org

0, 1, 8, 99, 1684, 36865, 994986, 32106655, 1209994808, 52281293697, 2551380861070, 138903509144191, 8350198884092484, 549502839975044449, 39295464010757324930, 3034457861009541582015, 251666093876245502584816, 22310882229970705663827457

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A052871, A376100.

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-x/(1-3*x)))))

a(n) = n!*sum(k=1, n, 3^(n-k)*k^(k-1)*binomial(n-1, k-1)/k!);

E.g.f. A(x) satisfies A(x) = x * (3*A(x) + exp(A(x))).
E.g.f.: Series_Reversion( x / (3*x + exp(x)) ).
a(n) = n! * Sum_{k=1..n} 3^(n-k) * k^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ (1 + 3*exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Sep 10 2024

A376123 E.g.f. A(x) satisfies A(x) = x * exp(A(x)) * (1 + 2*A(x)).

Original entry on oeis.org

0, 1, 6, 69, 1216, 29145, 886176, 32692597, 1419067392, 70867571409, 4002985561600, 252350116482981, 17564151708647424, 1337849793390444841, 110694246048458612736, 9886625352559043695125, 948044647019001482838016, 97146789899768662622795553

Offset: 0

Seiichi Manyama, Sep 11 2024

nonn

Index entries for reversions of series

Cf. A052885, A376124.

Cf. A088692, A376093, A376100.

a(n) = n!*sum(k=1, n, 2^(n-k)*n^(k-1)*binomial(n-1, k-1)/k!);

E.g.f.: Series_Reversion( x * exp(-x) / (1 + 2*x) ).
a(n) = n! * Sum_{k=1..n} 2^(n-k) * n^(k-1) * binomial(n-1,k-1)/k!.
a(n) = n * A088692(n-1).
a(n) ~ 2^(2*n) * n^(n-1) / (sqrt(3) * exp(n/2)). - Vaclav Kotesovec, Sep 11 2024

cp's OEIS Frontend

A376106 Expansion of e.g.f. LambertW(x / (1 - 2*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A376101 Expansion of e.g.f. -LambertW(-x / (1 - 3*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A376123 E.g.f. A(x) satisfies A(x) = x * exp(A(x)) * (1 + 2*A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula