A357243 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).
1, 1, -2, 6, -52, 540, -7608, 129304, -2612608, 60867360, -1608663840, 47527158624, -1552431588288, 55547889458880, -2160724031160576, 90782738645280000, -4097139872604807168, 197675862365363088384, -10153243488783257091072
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
nmax = 20; A[_] = 1; Do[A[x_] = ((1 - x)^(-1 + x))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-(1-x)*log(1-x))^k/k!))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-(1-x)*log(1-x))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1-x)*log(1-x)/lambertw(-(1-x)*log(1-x))))
Formula
E.g.f. satisfies A(x)^A(x) * (1 - x)^(1 - x) = 1.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-(1-x) * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-(1-x) * log(1-x)) ).
E.g.f.: A(x) = -(1-x) * log(1-x)/LambertW(-(1-x) * log(1-x)).