A277502 -id:A277502 - cp's OEIS frontend

A385424 Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).

1, 1, 3, 17, 137, 1465, 19499, 311873, 5829073, 124796081, 3012319315, 80960234577, 2398138520409, 77630951407529, 2726829925494011, 103300796618253825, 4198494172961579169, 182239547736082960737, 8414068749731088539299, 411754575622058760824593

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

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

Cf. A385426, A385427.

Cf. A277502, A381142, A385343, A385425.

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

E.g.f. A(x) satisfies A(x) = exp( arcsin(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A385343(n,k).
a(n) ~ n^(n-1) / (sqrt(cos(exp(-1))) * sin(exp(-1))^(n - 1/2) * exp(n - 3/2)). - Vaclav Kotesovec, Jun 28 2025

A277503 E.g.f.: -LambertW(-arctan(x)).

Original entry on oeis.org

0, 1, 2, 7, 48, 469, 5584, 80235, 1367040, 26840841, 595623680, 14752565807, 403579762688, 12084385256605, 393093330282496, 13804177426246995, 520496287836012544, 20973496057176404881, 899452315670554017792, 40903215737685386469847

Offset: 0

Vaclav Kotesovec, Oct 18 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A277502.

CoefficientList[Series[-LambertW[-ArcTan[x]], {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^50); concat([0], Vec(serlaplace(-lambertw(-atan(x))))) \\ G. C. Greubel, Nov 12 2017

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

cp's OEIS Frontend

A385424 Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).

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