A356909 E.g.f. satisfies A(x)^A(x) = 1/(1 - 2*x)^(1/2).

1, 1, 1, 6, 25, 260, 2031, 29414, 343729, 6331464, 99347775, 2213854962, 43775364777, 1143045179484, 27318538754703, 817488701386590, 22930078960458081, 773538392955105552, 24914603847636688383, 935548306987015373418, 34024565857537814925465

Offset: 0

Author

Seiichi Manyama, Sep 03 2022

Keywords

• nonn

Links

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

Crossrefs

Cf. A355786, A356908.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = (1 - 2*x)^(-(1/2)/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

a(n) = sum(k=0, n, 2^(n-k)*(-k+1)^(k-1)*abs(stirling(n, k, 1)));

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-log(1-2*x)/2)^k/k!)))

PARI

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

PARI

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

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * (-k+1)^(k-1) * |Stirling1(n,k)|.

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-log(1-2*x)/2)^k / k!.

E.g.f.: A(x) = exp( LambertW(-log(1-2*x)/2) ).

E.g.f.: A(x) = -log(1-2*x)/(2 * LambertW(-log(1-2*x)/2)).