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A369550
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Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).
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2
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1, 1, 3, 13, 85, 701, 6901, 79045, 1049385, 15924025, 271248121, 5108389001, 105158055949, 2346022349269, 56348945801877, 1449434215375021, 39758549273200081, 1159092552400164977, 35813081725133941297, 1169791166246561367697, 40297553373717279300981, 1460613225168596836153741
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OFFSET
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0,3
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COMMENTS
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Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).
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LINKS
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FORMULA
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E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = exp(x) * A(x^2*exp(x)).
(2) A(x) = exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = exp(L(x)) where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
(4) A(x) = G(x)/x where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
a(n) = A369090(n+1)/(n+1) for n >= 0.
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + 15924025*x^9/9! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
exp(-x) * A(x) = A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 380*x^5/5! + 3750*x^6/6! + + 42882*x^7/7! + 576296*x^8/8! + ...
The logarithm of e.g.f. A(x) equals L(x) where L(x) = x + L(x^2*exp(x)),
L(x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...
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PROG
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(PARI) {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1, n, A = exp(X) * subst(A, x, x^2*exp(X)) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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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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