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A209277
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O.g.f. satisfies: A(x) = Sum_{n>=0} (n+3)^n * x^n * A((n+3)*x)^n/n! * exp(-(n+3)*x*A((n+3)*x)).
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3
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1, 1, 8, 321, 57879, 45643415, 154158595175, 2190765237132015, 129241431881731600186, 31339180791153421540163500, 31011964321205837200260130287298, 124581202469689320858858825068619255535, 2023924731754579903607034623889070335771466703
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OFFSET
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0,3
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COMMENTS
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Compare to the LambertW identity:
Sum_{n>=0} (n+3)^n * x^n * G(x)^n/n! * exp(-(n+3)*x*G(x)) = 1/(1 - x*G(x)).
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LINKS
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EXAMPLE
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O.g.f.: A(x) = 1 + x + 8*x^2 + 321*x^3 + 57879*x^4 + 45643415*x^5 +...
where
A(x) = exp(-3*x*A(3*x)) + 4*x*A(4*x)*exp(-4*x*A(4*x)) + 5^2*x^2*A(5*x)^2/2!*exp(-5*x*A(5*x)) + 6^3*x^3*A(6*x)^3/3!*exp(-6*x*A(6*x)) + 7^4*x^4*A(7*x)^4/4!*exp(-7*x*A(7*x)) + 8^5*x^5*A(8*x)^5/5!*exp(-8*x*A(8*x)) +...
simplifies to a power series in x with integer coefficients.
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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, (k+3)^k*x^k*subst(A, x, (k+3)*x)^k/k!*exp(-(k+3)*x*subst(A, x, (k+3)*x)+x*O(x^n)))); polcoeff(A, n)}
for(n=0, 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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