OFFSET
1,3
FORMULA
Let A_n(x) denote the n-th iteration of A(x), then
. [A_{n+1}(x)]^3 = A(x)^3 * A_n'(x) for all n,
and A = A(x) satisfies:
. A = x + A^3 + A^3*D(A^3)/2! + A^3*D(A^3*D(A^3))/3! + A^3*D(A^3*D(A^3*D(A^3)))/4! + ...;
. A_n(x) = x + n*A^3 + n^2*A^3*D(A^3)/2! + n^3*A^3*D(A^3*D(A^3))/3! + n^4*A^3*D(A^3*D(A^3*D(A^3)))/4! + ...
where operator D(F) = d/dx F.
EXAMPLE
E.g.f: A(x) = x + 6*x^3/3! + 540*x^5/5! + 156240*x^7/7! + 96480720*x^9/9! + 104661849600*x^11/11! + 177947471782080*x^13/13! + 439942718370355200*x^15/15! +...
Related expansions:
A(x)^3 = 6*x^3/3! + 360*x^5/5! + 83160*x^7/7! + 43908480*x^9/9! +...
A(A(x)) = x + 12*x^3/3! + 1440*x^5/5! + 509040*x^7/7! + 368686080*x^9/9! +...
A(A(x))^3 = 6*x^3/3! + 720*x^5/5! + 241920*x^7/7! + 165110400*x^9/9! +...
A'(x) = 1 + 6*x^2/2! + 540*x^4/4! + 156240*x^6/6! + 96480720*x^8/8! +...
PROG
(PARI) /* Coefficients of A_m(x) = m-th iteration of A(x): */
{a(n, m=1)=local(A=x+x^3, D); for(i=1, n, D=x; A=x+sum(k=1, n, m^k*(D=(A+x*O(x^n))^3*deriv(D))/k!)); if(n<1, 0, n!*polcoeff(A, n))}
CROSSREFS
KEYWORD
eigen,nonn
AUTHOR
Paul D. Hanna, Aug 02 2010
STATUS
approved