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!)
A141372 G.f. satisfies: A(x) = x + A(A(A(x)))^2. 4
1, 1, 6, 57, 684, 9512, 146848, 2455208, 43764802, 822963750, 16203122280, 332189276516, 7062047285812, 155178233311932, 3515420453148936, 81936668615592785, 1961578144170589430, 48167700575393576406 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..18.

FORMULA

G.f. A(x) satisfies:

(1) A( x - A(A(x))^2 ) = x.

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(A(x))^(2*n) / n!.

(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(A(x))^(2*n)/x / n! ).

EXAMPLE

G.f.: A(x) = x + x^2 + 6*x^3 + 57*x^4 + 684*x^5 + 9512*x^6 +...

The g.f. satisfies the series:

A(x) = x + A(A(x))^2 + d/dx A(A(x))^4/2! + d^2/dx^2 A(A(x))^6/3! + d^3/dx^3 A(A(x))^8/4! +...

as well as the logarithmic series:

log(A(x)/x) = A(A(x))^2/x + [d/dx A(A(x))^4/x]/2! + [d^2/dx^2 A(A(x))^6/x]/3! + [d^3/dx^3 A(A(x))^8/x]/4! +...

Related expansions.

A(A(x)) = x + 2*x^2 + 14*x^3 + 145*x^4 + 1848*x^5 + 26920*x^6 +...

A(A(A(x))) = x + 3*x^2 + 24*x^3 + 270*x^4 + 3658*x^5 + 55970*x^6 +...

A(A(A(x)))^2 = x^2 + 6*x^3 + 57*x^4 + 684*x^5 + 9512*x^6 +...

The series reversion of A(x) = x - A(A(x))^2, where

A(A(x))^2 = x^2 + 4*x^3 + 32*x^4 + 346*x^5 + 4472*x^6 + 65292*x^7 +...

PROG

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+subst(A^2, x, subst(A, x, A+x*O(x^n)))); polcoeff(A, n)}

(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, subst(A, x, A+x*O(x^n))^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}

for(n=1, 25, print1(a(n), ", "))

(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, subst(A, x, A+x*O(x^n))^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}

for(n=1, 25, print1(a(n), ", "))

CROSSREFS

Cf. A141370, A141371.

Sequence in context: A324447 A060435 A153851 * A306030 A152170 A087659

Adjacent sequences:  A141369 A141370 A141371 * A141373 A141374 A141375

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 28 2008

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 November 30 08:16 EST 2021. Contains 349419 sequences. (Running on oeis4.)