OFFSET
0,3
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^2 satisfies the following formulas.
(1) A(x) = ( Sum_{n>=0} x^n/n!^2 )^A(x).
(2) A(x) = log(A(x)) / ( Sum_{n>=0} A101981(n)*x^n/n!^2 ).
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2!^2 + 73*x^3/3!^2 + 2061*x^4/4!^2 + 97301*x^5/5!^2 +...
where
A(x) = (1 + x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 + x^5/5!^2 +...)^A(x).
Related expansions:
log(A(x)) = x + 3*x^2/2!^2 + 31*x^3/3!^2 + 679*x^4/4!^2 + 25581*x^5/5!^2 + 1474706*x^6/6!^2 + 120670201*x^7/7!^2 + 13298986863*x^8/8!^2 +...
log(A(x))/A(x) = log(1 + x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 +...);
log(A(x))/A(x) = x - x^2/2!^2 + 4*x^3/3!^2 - 33*x^4/4!^2 + 456*x^5/5!^2 - 9460*x^6/6!^2 + 274800*x^7/7!^2 +...+ A101981(n)*x^n/n!^2 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/m!^2+x*O(x^n))^A); n!^2*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 07 2012
STATUS
approved