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A349652
E.g.f. satisfies: A(x)^(A(x)^2) = 1/(1 - x).
6
1, 1, -2, 18, -220, 3880, -86868, 2376836, -76735216, 2856604464, -120457684320, 5675047644288, -295430737430112, 16840861797433440, -1043322313406139648, 69798144929293516800, -5014888682767294232832, 385130588783629323238656
OFFSET
0,3
LINKS
Eric Weisstein's World of Mathematics, Lambert W-Function.
FORMULA
a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = -Sum_{k>=0} (2*k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = ( -2*log(1-x)/LambertW(-2*log(1-x)) )^(1/2).
a(n) ~ -(-1)^n * n^(n-1) / (sqrt(2) * exp(n + exp(-1)/4) * (-1 + exp(exp(-1)/2))^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021
MATHEMATICA
nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1 - x))^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
PROG
(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*stirling(n, k, 1));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (2*k-1)^(k-1)*log(1-x)^k/k!)))
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*log(1-x)/lambertw(-2*log(1-x)))^(1/2)))
CROSSREFS
KEYWORD
sign
AUTHOR
Seiichi Manyama, Nov 23 2021
STATUS
approved