OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A(x - x^2 - A(x)^3).
(2) A(x) = x + A(x)^2 + A(A(x))^3.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (x^2 + A(x)^3)^n / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (x^2 + A(x)^3)^n/x / n! ).
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 13*x^4 + 68*x^5 + 401*x^6 + 2576*x^7 + 17670*x^8 + 127786*x^9 + 965890*x^10 + 7583944*x^11 + 61576198*x^12 + ...
where A(x - x^2 - A(x)^3) = x.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 58*x^6 + 318*x^7 + 1911*x^8 + 12330*x^9 + 84273*x^10 + 604503*x^11 + ...
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 171*x^6 + 1016*x^7 + 6531*x^8 + 44666*x^9 + 321418*x^10 + ...
A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 230*x^6 + 1560*x^7 + 11139*x^8 + 83120*x^9 + 644472*x^10 + ...
where A(A(x))^3 = A(x) - x - A(x)^2.
A(A(x)) = x + 2*x^2 + 8*x^3 + 42*x^4 + 256*x^5 + 1721*x^6 + 12424*x^7 + 94796*x^8 + 756680*x^9 + ...
where A(A(y)) = x at y = x - 2*x^2 + x^3 - x^4 - (1 - 2*x + 2*x^2)*A(x)^3 - A(x)^6.
PROG
(PARI) {a(n) = my(A=x); if(n<1, 0, for(i=1, n, A = serreverse(x - x^2 - A^3 + x*O(x^n))); polcoeff(A, n))}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (x^2 + A^3)^m)/m! +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, (x^2 + A^3)^m/x)/m!) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 20 2024
STATUS
approved