A357247 E.g.f. satisfies A(x) * log(A(x)) = x * exp(-x).
1, 1, -3, 13, -103, 1241, -19691, 384805, -8918351, 238966705, -7265920339, 247123552061, -9295263915191, 383095792217737, -17167554097899323, 831082449069928021, -43221681697593767071, 2403219105771778162529, -142263939562414917333155
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..372
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Maple
A357247 := n -> (-1)^(n - 1) * add(binomial(n, j) * (j - 1)^(j - 1) * j^(n - j), j = 0..n): seq(A357247(n), n = 0..18); # Peter Luschny, Jan 28 2023
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Mathematica
nmax = 20; A[_] = 1; Do[A[x_] = Exp[x/(Exp[x]*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)*(x*exp(-x))^k/k!))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*exp(-x))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(x*exp(-x)/lambertw(x*exp(-x)))) -
PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=1, j, (-k)^(j-1)*binomial(j, k))*binomial(i-1, j-1)*v[i-j+1])); v;
Formula
E.g.f. satisfies A(x) * log(A(x)) - x * exp(-x) = 0.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * exp(-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(x * exp(-x)) ).
E.g.f.: A(x) = x * exp(-x)/LambertW(x * exp(-x)).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * A216857(k) * binomial(n-1,k-1) * a(n-k).
a(n) = (-1)^(n - 1) * Sum_{j=0..n} binomial(n, j) * (j - 1)^(j - 1) * j^(n - j). - Peter Luschny, Jan 28 2023
a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n+1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Jan 28 2023