A141372 G.f. satisfies: A(x) = x + A(A(A(x)))^2.

1, 1, 6, 57, 684, 9512, 146848, 2455208, 43764802, 822963750, 16203122280, 332189276516, 7062047285812, 155178233311932, 3515420453148936, 81936668615592785, 1961578144170589430, 48167700575393576406

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