A376106 -id:A376106 - cp's OEIS frontend

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

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!.

A377373 Expansion of e.g.f. (1/x) * Series_Reversion( x / (exp(-x) + 2*x) ).

Original entry on oeis.org

1, 1, 3, 14, 93, 794, 8335, 103774, 1496313, 24525458, 450478131, 9166307798, 204692557333, 4977320639290, 130918278855351, 3703846153114574, 112155490349101041, 3619411771703973410, 124011196515200953819, 4496024219722304736070, 171963129575721708667341

Offset: 0

Seiichi Manyama, Dec 28 2024

nonn

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

Cf. A108919, A377374.

Cf. A376106.

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

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

E.g.f.: (1/x) * LambertW(x / (1 - 2*x)).
a(n) = n! * Sum_{k=0..n} (-1)^k * 2^(n-k) * (k+1)^(k-1) * binomial(n,k)/k!.
a(n) = A376106(n+1)/(n+1).

A376327 Expansion of e.g.f. -LambertW(-x/(1-x^2)).

Original entry on oeis.org

0, 1, 2, 15, 112, 1285, 17616, 299299, 5946368, 136497897, 3544641280, 102858065431, 3297199331328, 115730076038317, 4414151526557696, 181797547951527915, 8040649885153755136, 380100842138029431121, 19125314442962053300224, 1020539634854353310016415, 57563650890815727219507200

Offset: 0

Vaclav Kotesovec, Sep 20 2024

nonn

Cf. A052871, A376106.

nmax=25; CoefficientList[Series[-LambertW[-x/(1-x^2)], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) ~ (1 + 4*exp(-2))^(1/4) * 2^n * n^(n-1) / (exp(n) * (sqrt(4 + exp(2)) - exp(1))^n).

cp's OEIS Frontend

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

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A377373 Expansion of e.g.f. (1/x) * Series_Reversion( x / (exp(-x) + 2*x) ).

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A376327 Expansion of e.g.f. -LambertW(-x/(1-x^2)).

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