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A259267
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E.g.f. A(x) satisfies: A'(x) = exp(2*A(A(x))).
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2
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1, 2, 12, 128, 2016, 42656, 1145280, 37563008, 1464675840, 66533778944, 3466031815680, 204489094565888, 13524452573872128, 994257291909816320, 80668058806271016960, 7179145234347383128064, 697131195162680465817600, 73522035747248454761578496, 8387016414085244676889116672
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OFFSET
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1,2
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) A''(x) = 2*exp( 4*A(A(x)) + 2*A(A(A(x))) ).
(2) exp(-2*A(x)) = d/dx Series_Reversion(A(x)).
(3) A(x) = log(F(x)) where F(x) satisfies: F( Integral 1/F(x)^2 dx ) = exp(x) and equals the e.g.f. of A233336.
a(n) = 2^(n-1) * A214645(n) for n>=1.
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EXAMPLE
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E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! +...
Related expansions:
A'(x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2016*x^4/4! + 42656*x^5/5! +...
A(A(x)) = log(A'(x))/2 = x + 4*x^2/2! + 36*x^3/3! + 520*x^4/4! + 10512*x^5/5! + 276064*x^6/6! + 8987712*x^7/7! + 351278080*x^8/8! +...
The exponential of e.g.f. A(x) equals the e.g.f. of A233336:
exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! + 1677883*x^7/7! +...+ A233336(n)*x^n/n! +...
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PROG
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(PARI) {a(n)=local(A=x+x^2); for(i=0, n, A=intformal(exp(2*subst(A, x, A+x*O(x^n))))); n!*polcoeff(A, n)}
for(n=1, 25, 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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