OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..136
FORMULA
E.g.f. satisfies: A(x) = exp( x*A(log A(x)) ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1).
E.g.f: A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the e.g.f. of A144682.
E.g.f: A(x) = G(3x)^(1/3) where G(x/G(x)^3) = exp(x) and G(x) is the e.g.f. of A144683.
E.g.f: A(x) = G(4x)^(1/4) where G(x/G(x)^4) = exp(x) and G(x) is the e.g.f. of A144684.
E.g.f: A(x) = 1/G(-x) where G(x*G(x)) = exp(x) and G(x) is the e.g.f. of A087961.
E.g.f. A(log(A(x))) = log(A(x))/x = G(x) is the e.g.f of A140049 where G(x) satisfies G(x*exp(-x*G(x))) = exp(x*G(x)).
From Seiichi Manyama, Jun 10 2026: (Start)
E.g.f.: exp(B(x)), where B(x) is the e.g.f. of A140054.
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n-1,k) * a(k) * A140054(n-k). (End)
EXAMPLE
E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...
A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...
1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...
A(log(A(x))) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:
exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];
exp(x)*A(x)^2: [ 1, (3), 13, 93, 1145, 22593, 645741, ...];
exp(x)*A(x)^3: [ 1, 4, (22), 181, 2320, 45199, 1257364, ...];
exp(x)*A(x)^4: [ 1, 5, 33, (305), 4097, 79825, 2177329, ...];
exp(x)*A(x)^5: [ 1, 6, 46, 471, (6656), 131001, 3529836, ...];
exp(x)*A(x)^6: [ 1, 7, 61, 685, 10201, (204337), 5477005, ...];
exp(x)*A(x)^7: [ 1, 8, 78, 953, 14960, 306643, (8226436), ...];
...
then the terms along the main diagonal form this sequence shift left.
PROG
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(n=0, n, A=exp(serreverse(x/A))); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^n, n-1))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 19 2008
STATUS
approved