A179936 E.g.f. satisfies: A(A(x))^3 = A(x)^3 * A'(x).

1, 0, 6, 0, 540, 0, 156240, 0, 96480720, 0, 104661849600, 0, 177947471782080, 0, 439942718370355200, 0, 1499983925770422892800, 0, 6780179728814814933811200, 0, 39412342523061395825857920000, 0

Offset: 1

Author

Paul D. Hanna, Aug 02 2010

Keywords

• eigen
• nonn

Examples

			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! +...
		

Crossrefs

Cf. A179497 (variant).

Programs

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))}

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.