A141370 G.f. satisfies: A(x) = x + A(A(A(x)^2)).

1, 1, 2, 7, 26, 104, 436, 1894, 8444, 38418, 177656, 832548, 3945156, 18871524, 91003360, 441927367, 2159282462, 10607708284, 52363342484, 259601860898, 1292041756732, 6453179670344, 32334136480656, 162487089008766

Offset

1,3

Links

Table of n, a(n) for n=1..24.

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 + 2*x^3 + 7*x^4 + 26*x^5 + 104*x^6 + 436*x^7 +...
The series reversion of A(x) = x - A(A(x^2)), where
A(A(x^2)) = x^2 + 2*x^4 + 6*x^6 + 25*x^8 + 116*x^10 + 574*x^12 + 2972*x^14 +...
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, x, A^2+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, subst(A, x, x^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, subst(A, x, x^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, subst(A, x, x^2+x*O(x^n)))^m/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A141371, A141372.

Sequence in context: A150545 A150546 A151296 * A150547 A150548 A150549

Adjacent sequences: A141367 A141368 A141369 * A141371 A141372 A141373

Keyword

nonn

Author

Paul D. Hanna, Jun 28 2008

Status

approved