%I #11 Oct 02 2024 07:54:20

%S 1,1,3,22,305,6656,204337,8226436,414585425,25315924960,1828704716801,

%T 153433983789164,14739472821255481,1602471473448455104,

%U 195300935112810494801,26470100501608768436716

%N E.g.f. satisfies: A(x/A(x)) = exp(x).

%H Vaclav Kotesovec, <a href="/A144681/b144681.txt">Table of n, a(n) for n = 0..136</a>

%F E.g.f. satisfies: A(x) = exp( x*A(log A(x)) ).

%F E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1).

%F 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.

%F 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.

%F 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.

%F 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.

%F 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)).

%e E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...

%e A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...

%e 1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...

%e A(log(A(x))) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...

%e ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.

%e Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:

%e exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];

%e exp(x)*A(x)^2: [1,(3), 13, 93, 1145, 22593, 645741, ...];

%e exp(x)*A(x)^3: [1, 4,(22), 181, 2320, 45199, 1257364, ...];

%e exp(x)*A(x)^4: [1, 5, 33,(305), 4097, 79825, 2177329, ...];

%e exp(x)*A(x)^5: [1, 6, 46, 471,(6656), 131001, 3529836, ...];

%e exp(x)*A(x)^6: [1, 7, 61, 685, 10201,(204337), 5477005, ...];

%e exp(x)*A(x)^7: [1, 8, 78, 953, 14960, 306643,(8226436), ...]; ...

%e then the terms along the main diagonal form this sequence shift left.

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A)));n!*polcoeff(A,n)}

%o (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))}

%Y Cf. A144682, A144683, A144684, A087961, A140049.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 19 2008