A193202 E.g.f. A(x) satisfies: A(A(x)) = x*A'(A(x)).

1, 2, 0, 12, -160, 3240, -86688, 2922640, -119971584, 5847901920, -332122243200, 21653202377664, -1601381638172160, 133036354347921024, -12314128238585510400, 1261212911036957548800, -142082122642808666185728, 17514853400850824425213440, -2351847513553411263501035520, 342599734607249938595012582400

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..150

Formula

E.g.f. satisfies: A( A(x)/A'(x) ) = x.

E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) = -A(-x)/A'(-x) is the e.g.f. of A179420 and satisfies: G(G(x)) = x*G'(x).

The inverse function of A(x), A(x)/A'(x), equals the g.f. of column 0 of the matrix log of the Riordan array (A(x)/x, A(x)).

Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x, then:

_ A(x)/A'(x) = A_{n-1}(x) * A_n(x) / (x * d/dx A_n(x)) for all n.

Example

E.g.f.: A(x) = x + 2*x^2/2! + 12*x^4/4! - 160*x^5/5! + 3240*x^6/6! - 86688*x^7/7! + 2922640*x^8/8! - 119971584*x^9/9! + 5847901920*x^10/10! +...
RELATED EXPANSIONS.
_ A(A(x)) = x + 4*x^2/2! + 12*x^3/3! + 48*x^4/4! + 40*x^5/5! + 2640*x^6/6! - 57456*x^7/7! + 2059904*x^8/8! - 85967136*x^9/9! + 4262310720*x^10/10! +...
_ A'(A(x)) = 1 + 2*x + 4*x^2/2! + 12*x^3/3! + 8*x^4/4! + 440*x^5/5! - 8208*x^6/6! +...
_ A(x)/A'(x) = x - 2*x^2/2! + 12*x^3/3! - 132*x^4/4! + 2200*x^5/5! - 50280*x^6/6! + 1482768*x^7/7! - 54171376*x^8/8! + 2381590944*x^9/9! - 123292821600*x^10/10! +...
Higher order iterations begin:
_ A_3(x) = x + 6*x^2/2! + 36*x^3/3! + 252*x^4/4! + 1800*x^5/5! + 16920*x^6/6! +...
_ A_4(x) = x + 8*x^2/2! + 72*x^3/3! + 768*x^4/4! + 9200*x^5/5! + 126720*x^6/6! +...
_ A_5(x) = x + 10*x^2/2! + 120*x^3/3! + 1740*x^4/4! + 29200*x^5/5! + 561000*x^6/6! +...
Illustrate a main property of the iterations A_n(x) by:
_ A(x)/A'(x) = A(x) * A(A(x)) / (x*d/dx A(A(x)));
_ A(x)/A'(x) = A_2(x) * A_3(x) / (x*d/dx A_3(x));
_ A(x)/A'(x) = A_3(x) * A_4(x) / (x*d/dx A_4(x));
_ A(x)/A'(x) = A_4(x) * A_5(x) / (x*d/dx A_5(x)); ...
which can be shown consistent by the chain rule of differentiation.

Prog

(PARI) {a(n)=local(A=x+x^2+sum(m=3, n-1, a(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n), n!*polcoeff(subst(A, x, A)-x*subst(A', x, A), n)/(n-2))}

Crossrefs

Cf. A179420.

Sequence in context: A293567 A293494 A058803 * A294463 A107700 A274107

Adjacent sequences: A193199 A193200 A193201 * A193203 A193204 A193205

Keyword

sign

Author

Paul D. Hanna, Jul 22 2011

Status

approved