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A307615
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E.g.f. A(x) satisfies: A(x) = exp(x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...
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2
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1, 1, 5, 31, 337, 2741, 40621, 474475, 8461601, 132034537, 2648537461, 50079699671, 1204884343345, 26450428964701, 697107087763997, 17873985363570211, 526080367468142401, 15060611189639187665, 487251625325328212581, 15494976568071805188367, 545902629556769672596241
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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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E.g.f.: exp(Sum_{k>=1} A050369(k)*x^k).
a(0) = 1; a(n) = Sum_{k=1..n} A074206(k)*k*k!*binomial(n-1,k-1)*a(n-k).
a(n) ~ (-Gamma(2+r)/zeta'(r))^(1/(4 + 2*r)) * exp(-n + 12/25 + n^(1 - 1/(2+r)) * (2+r) * (-Gamma(2+r)/zeta'(r))^(1/(2+r)) / (1+r)) * n^(n - 1/(4 + 2*r)) / sqrt(2+r), where r = A107311 = 1.7286472389981836181351... is the root of the equation Zeta(r) = 2, Zeta'(r) = -1/A247667. - Vaclav Kotesovec, Aug 09 2021
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 5*x^2/2! + 31*x^3/3! + 337*x^4/4! + 2741*x^5/5! + 40621*x^6/6! + 474475*x^7/7! + 8461601*x^8/8! + 132034537*x^9/9! + ...
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MATHEMATICA
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terms = 20; A[_] = 1; Do[A[x_] = Exp[x] Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x] Range[0, terms]!
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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