A141371 G.f. satisfies: A(x) = x + A(A(A(x))^2).

1, 1, 4, 25, 190, 1648, 15700, 160834, 1747372, 19945264, 237585064, 2938737760, 37602695500, 496263658816, 6739460289568, 94002095361937, 1344557294558722, 19696746902333368, 295199862739677892, 4522172757314573464

Offset

1,3

Links

Table of n, a(n) for n=1..20.

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 + 4*x^3 + 25*x^4 + 190*x^5 + 1648*x^6 + 15700*x^7 +...
The series reversion of A(x) = x - A(A(x)^2), where
A(A(x)^2) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 472*x^6 + 4052*x^7 + 38227*x^8 +...
The g.f. satisfies the series:
A(x) = x + A(A(x)^2) + d/dx A(A(x)^2)^2/2! + d^2/dx^2 A(A(x)^2)^3/3! + d^3/dx^3 A(A(x)^2)^4/4! +...
as well as the logarithmic series:
log(A(x)/x) = A(A(x)^2)/x + [d/dx A(A(x)^2)^2/x]/2! + [d^2/dx^2 A(A(x)^2)^3/x]/3! + [d^3/dx^3 A(A(x)^2)^4/x]/4! +...

Prog

(PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x+subst(A, x, subst(A^2, x, A+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - subst(A, x, A^2+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+sum(m=1, n, Dx(m-1, subst(A, x, A^2+x*O(x^n))^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^2+x*O(x^n))^m/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A141370, A141372.

Sequence in context: A239996 A064063 A171991 * A224079 A199494 A171791

Adjacent sequences: A141368 A141369 A141370 * A141372 A141373 A141374

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2008

Status

approved