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A216247
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O.g.f. satisfies: A(x) = Sum_{n>=0} n^n * x^n * A(n^2*x^2)^n/n! * exp(-n*x*A(n^2*x^2)).
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1
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1, 1, 1, 2, 15, 77, 354, 2509, 19369, 146840, 1321773, 13466872, 151177886, 1827144180, 23686530621, 335038346222, 4922031402993, 75768551557262, 1240672912146220, 21264349760865217, 384135346377854363, 7286741020161077248, 145190876484745421540
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OFFSET
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0,4
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COMMENTS
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Compare to the LambertW identity:
Sum_{n>=0} n^n * x^n * G(x)^n/n! * exp(-n*x*G(x)) = 1/(1 - x*G(x)).
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LINKS
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EXAMPLE
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G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 15*x^4 + 77*x^5 + 354*x^6 + 2509*x^7 +...
where
A(x) = 1 + x*A(x^2)*exp(-x*A(x^2)) + 2^2*x^2*A(2^2*x^2)^2/2!*exp(-2*x*A(2^2*x^2)) + 3^3*x^3*A(3^2*x^2)^3/3!*exp(-3*x*A(3^2*x^2)) + 4^4*x^4*A(4^2*x^2)^4/4!*exp(-4*x*A(4^2*x^2)) + 5^5*x^5*A(5^2*x^2)^5/5!*exp(-5*x*A(5^2*x^2)) +...
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^k*x^k*subst(A, x, k^2*x^2)^k/k!*exp(-k*x*subst(A, x, k^2*x^2)+x*O(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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