login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A193202 E.g.f. A(x) satisfies: A(A(x)) = x*A'(A(x)). 2
1, 2, 0, 12, -160, 3240, -86688, 2922640, -119971584, 5847901920, -332122243200, 21653202377664, -1601381638172160, 133036354347921024, -12314128238585510400, 1261212911036957548800, -142082122642808666185728, 17514853400850824425213440, -2351847513553411263501035520, 342599734607249938595012582400 (list; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 17:01 EST 2021. Contains 349596 sequences. (Running on oeis4.)