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A203013
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Expansion of e.g.f. A(x) satisfying A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.
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7
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1, 1, 5, 55, 993, 24871, 802873, 31793035, 1493163745, 81186783535, 5018214016041, 347636382949747, 26685235607680081, 2248760378885064487, 206430769607981879353, 20507793044444903462251, 2192507508237447321800385, 251034864831917236610746207
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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. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following.
(1) A(x) = Sum_{n>=0} (2*A(x)^n - 1)^n * x^n/n!.
(2) A(x) = Sum_{n>=0} 2^n * A(x)^(n^2) * exp(-x*A(x)^n) * x^n/n!.
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 5*x^2/2! + 55*x^3/3! + 993*x^4/4! + 24871*x^5/5! +...
where the e.g.f. satisfies following series identity:
A(x) = 1 + (2*A(x)-1)*x + (2*A(x)^2-1)^2*x^2/2! + (2*A(x)^3-1)^3*x^3/3! + (2*A(x)^4-1)^4*x^4/4! +...
is equal to
A(x) = exp(-x) + 2*A(x)*exp(-x*A(x))*x + 2^2*A(x)^4*exp(-x*A(x)^2)*x^2/2! + 2^3*A(x)^9*exp(-x*A(x)^3)*x^3/3! + 2^4*A(x)^16*exp(-x*A(x)^4)*x^4/4! +...
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, (2*A^k-1+x*O(x^n))^k*x^k/k!)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, 2^k*A^(k^2)*exp(-A^k*x+x*O(x^n))*x^k/k!)); n!*polcoeff(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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