A376101 -id:A376101 - cp's OEIS frontend

A376100 Expansion of e.g.f. -LambertW(-x / (1 - 2*x)).

0, 1, 6, 57, 760, 13265, 289116, 7600873, 234730224, 8340307137, 335388171700, 15062758093361, 747393408423432, 40606032733746961, 2397539426985311532, 152864047998089113785, 10467226142002168282336, 766094017043351707135745

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A052871, A376101.

my(N=20, 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!.
a(n) ~ (1 + 2*exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Sep 10 2024

A376107 Expansion of e.g.f. LambertW(x / (1 - 3*x)).

Original entry on oeis.org

0, 1, 4, 27, 260, 3265, 50634, 935263, 20053816, 489677697, 13416375950, 407609962111, 13600700469828, 494442286466401, 19452778285314178, 823489845351967935, 37323572563440199664, 1803303384581598518785, 92523649833821902792086

Offset: 0

Seiichi Manyama, Sep 10 2024

nonn

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

Cf. A060356, A376106.

Cf. A376101.

nmax=20; CoefficientList[InverseSeries[Series[x / (3*x + E^(-x)), {x, 0, nmax}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 20 2024 *)

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!.

A376124 E.g.f. A(x) satisfies A(x) = x * exp(A(x)) * (1 + 3*A(x)).

Original entry on oeis.org

0, 1, 8, 117, 2584, 77245, 2923776, 134147881, 7237959296, 449153207001, 31518672486400, 2468063119978621, 213353171897306112, 20181854221412587093, 2073632053555295657984, 229977288184566078752625, 27382787929802599713243136, 3483970698377702965413968689

Offset: 0

Seiichi Manyama, Sep 11 2024

nonn

Index entries for reversions of series

Cf. A052885, A376123.

Cf. A376094, A376101.

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

E.g.f.: Series_Reversion( x * exp(-x) / (1 + 3*x) ).
a(n) = n! * Sum_{k=1..n} 3^(n-k) * n^(k-1) * binomial(n-1,k-1)/k!.
a(n) ~ ((7 + sqrt(13))/2)^n * n^(n-1) / (13^(1/4) * exp((7 - sqrt(13))*n/6)). - Vaclav Kotesovec, Sep 11 2024

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A376100 Expansion of e.g.f. -LambertW(-x / (1 - 2*x)).

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A376107 Expansion of e.g.f. LambertW(x / (1 - 3*x)).

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A376124 E.g.f. A(x) satisfies A(x) = x * exp(A(x)) * (1 + 3*A(x)).

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