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| NAME
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allocated for Paul D. Hanna
G.f. satisfies: A(x) = Sum_{n>=0} log(1 + 2^n*x*A(x))^n/n!.
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| DATA
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1, 2, 10, 100, 2500, 224728, 77611032, 95603336016, 411188458873152, 6215509773143124736, 334390128406134844422816, 64839530694681966290325813952, 45813418110052719651124682371286592
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| OFFSET
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0,2
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| FORMULA
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G.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} C(2^n,n)*x^n*A(x)^n,
(2) A(x) = (1/x)*Series_Reversion(x/B(x)),
(3) A(x) = B(x*A(x)) and B(x) = A(x/B(x)),
where B(x) = Sum_{n>=0} C(2^n,n)*x^n is the g.f. of A014070.
(4) A(x) = G(x/A(x)) and G(x) = A(x*G(x)), where G(x) is the g.f. of A188194.
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| EXAMPLE
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G.f.: A(x) = 1 + 2*x + 10*x^2 + 100*x^3 + 2500*x^4 + 224728*x^5 +...
which equals the series:
A(x) = 1 + log(1+2*x*A(x)) + log(1+4*x*A(x))^2/2! + log(1+8*x*A(x))^3/3! +...
Let B(x) equal the g.f. of A014070, which begins:
B(x) = 1 + 2*x + 6*x^2 + 56*x^3 + 1820*x^4 +...+ C(2^n,n)*x^n +...
then B(x) = A(x/B(x)) and A(x) = B(x*A(x)), so that:
A(x) = 1 + 2*x*A(x) + 6*x^2*A(x)^2 + 56*x^3*A(x)^3 + 1820*x^4*A(x)^4 +...+ C(2^n,n)*x^n*A(x)^n +...
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| PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, log(1+2^m*x*A+x*O(x^n))^m/m!)); polcoeff(A, n)}
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| CROSSREFS
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Cf. A014070, A188194.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Mar 23 2011
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| STATUS
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approved
proposed
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