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A219265 O.g.f. satisfies: A(x) = Sum_{n>=0} A(n^2*x)^n * (n^2*x)^n/n! * exp(-n^2*x*A(n^2*x)). 2
1, 1, 8, 160, 6918, 609469, 106947753, 37651271215, 26931993643529, 39243099256414069, 116654228928308598913, 710224935200206160129234, 8867331728829780268501045551, 227187317486051730833557991305666, 11969414396907448200529521385052444890 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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)).

LINKS

Table of n, a(n) for n=0..14.

EXAMPLE

O.g.f.: A(x) = 1 + x + 8*x^2 + 160*x^3 + 6918*x^4 + 609469*x^5 +...

where

A(x) = 1 + x*A(x)*exp(-x*A(x)) + 2^4*x^2*A(2^2*x)^2/2!*exp(-2^2*x*A(2^2*x)) + 3^6*x^3*A(3^2*x)^3/3!*exp(-3^2*x*A(3^2*x)) + 4^8*x^4*A(4^2*x)^4/4!*exp(-4^2*x*A(4^2*x)) + 5^10*x^5*A(5^2*x)^5/5!*exp(-5^2*x*A(5^2*x)) +...

simplifies to a power series in x with integer coefficients.

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, k^(2*k)*x^k*subst(A, x, k^2*x)^k/k!*exp(-k^2*x*subst(A, x, k^2*x)+x*O(x^n)))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A219264, A218681, A218672, A219184, A217900.

Sequence in context: A036909 A221077 A052140 * A300466 A184605 A195607

Adjacent sequences:  A219262 A219263 A219264 * A219266 A219267 A219268

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 16 2012

STATUS

approved

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Last modified October 3 22:17 EDT 2022. Contains 357237 sequences. (Running on oeis4.)