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A349650
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E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x.
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6
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1, 1, -4, 36, -532, 11040, -295188, 9655772, -373422320, 16666348464, -843095987520, 47669276120928, -2979044176833888, 203906085094788960, -15170476121142482112, 1218972837861962011200, -105202043767190506428672, 9705514148732971389369600
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OFFSET
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0,3
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LINKS
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FORMULA
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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) * exp(n*(exp(-1)/2 - 1)) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021
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MATHEMATICA
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nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
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PROG
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(PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*abs(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)))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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