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A349657
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E.g.f. satisfies: A(x)^3 * log(A(x)) = 1 - exp(-x).
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5
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1, 1, -6, 80, -1751, 53402, -2088528, 99680667, -5617170700, 365003288652, -26868393676609, 2209797209486528, -200828403704351068, 19986049281174575497, -2161617056877509895386, 252467067400866652634004, -31668302130310076212791823
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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} (3*k-1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(3*(1 - exp(-x)))/3 ).
G.f.: Sum_{k>=0} (-3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ -(-1)^n * sqrt(3*exp(1) + 1) * sqrt(-log(3) + log(3 + exp(-1))) * n^(n-1) / (3 * exp(n + 1/3) * (-log(3) + log(3*exp(1) + 1) - 1)^n). - Vaclav Kotesovec, Nov 24 2021
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MATHEMATICA
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nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(Exp[x]-1)/(Exp[x]*A[x]^3)]+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, (3*k-1)^(k-1)*stirling(n, k, 2));
(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(1-exp(-x)))/3)))
(PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-3*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))
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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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