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

1,3

Links

Table of n, a(n) for n=1..22.

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

Cf. A179497 (variant).

Sequence in context: A336303 A111372 A294686 * A219952 A156444 A218343

Adjacent sequences: A179933 A179934 A179935 * A179937 A179938 A179939

Keyword

eigen,nonn

Author

Paul D. Hanna, Aug 02 2010

Status

approved