A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.
1, 0, 2, 3, -4, -30, 294, 3780, -7904, -444528, 78840, 99657360, 539299848, -27852945120, -361237078944, 10124338180320, 258341121976320, -4020500134465920, -205187357182405824, 1330097523844832640, 186823640933648588160, 500469438126120583680
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
nmax = 21; A[_] = 1; Do[A[x_] = (1/(1 - x)^x)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}]; CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *) -
PARI
a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x*log(1-x))^k/k!))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x*log(1-x))))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*log(1-x)/lambertw(-x*log(1-x))))
Formula
a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x * log(1-x)) ).
E.g.f.: A(x) = -x * log(1-x)/LambertW(-x * log(1-x)).