A362656 -id:A362656 - cp's OEIS frontend

A362671 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^2 ).

1, 1, -1, 10, -111, 1716, -33755, 807738, -22782207, 740204776, -27226430739, 1118416240470, -50750734988063, 2521219487859372, -136098630522431499, 7932551567421395866, -496501182232557828735, 33214032504796887027408

Offset: 0

Seiichi Manyama, Apr 29 2023

sign

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

Cf. A273954, A360473, A362656, A362672.

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

E.g.f.: exp( LambertW(2*x * exp(x))/2 ).

a(n) = Sum_{k=0..n} k^(n-k) * (-2*k+1)^(k-1) * binomial(n,k).

A362672 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^3 ).

Original entry on oeis.org

1, 1, -3, 37, -679, 17161, -553451, 21731053, -1006118863, 53671172113, -3241671266899, 218677223408821, -16296163119155063, 1329568681331536153, -117874745761237043515, 11283758432396431997821, -1159952212029532257663391, 127445385808282289840496673

Offset: 0

Seiichi Manyama, Apr 29 2023

sign

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

Cf. A273954, A360473, A362656, A362671.

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

E.g.f.: exp( LambertW(3*x * exp(x))/3 ).

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

A380881 E.g.f. A(x) satisfies A(x) = exp( x * A(x)^3 * exp(x * A(x)^3) ).

Original entry on oeis.org

1, 1, 9, 163, 4541, 171781, 8231395, 478055299, 32642065433, 2562896897353, 227510655792191, 22533214047347455, 2463465770439307045, 294676777871863052173, 38284087227668033391515, 5368383942726216941810971, 808133883137288259018215345, 129988823008132636178027546257

Offset: 0

Seiichi Manyama, Feb 07 2025

nonn

Index entries for reversions of series

Cf. A162695, A360474.

Cf. A362656, A380880.

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

E.g.f.: ( (1/x) * Series_Reversion( x * exp(-3*x*exp(x)) ) )^(1/3).

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

cp's OEIS Frontend

A362671 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^2 ).

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A362672 E.g.f. satisfies A(x) = exp( x * exp(x) / A(x)^3 ).

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A380881 E.g.f. A(x) satisfies A(x) = exp( x * A(x)^3 * exp(x * A(x)^3) ).

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