A144682 E.g.f. satisfies: A(x/A(x)^2) = exp(x).

1, 1, 5, 73, 2073, 92481, 5775133, 471058953, 47961475313, 5904863932609, 858738633997941, 144899744540718729, 27970301202134146441, 6106540658691499524993, 1493749158085983126737165

Offset

0,3

Links

Table of n, a(n) for n=0..14.

Formula

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^2 ).

E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2).

E.g.f. A(x) = G(x/2)^2 where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

Example

E.g.f. A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2073*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(2*n) for n>=1, k>=0:
exp(x)*A(x)^2: [(1), 3, 17, 219, 5665, 239283, 14432433, ...];
exp(x)*A(x)^4: [1,(5), 41, 605, 15633, 638325, 37250233, ...];
exp(x)*A(x)^6: [1, 7,(73), 1207, 31825, 1274407, 72322201, ...];
exp(x)*A(x)^8: [1, 9, 113,(2073), 56545, 2249769, 124959057, ...];
exp(x)*A(x)^10:[1, 11, 161, 3251,(92481), 3695451, 202282081, ...];
exp(x)*A(x)^12:[1, 13, 217, 4789, 142705,(5775133), 313637833, ...];
exp(x)*A(x)^14:[1, 15, 281, 6735, 210673, 8688975,(471058953), ...]; ...
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^2))); n!*polcoeff(A, n)}
{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^(2*n), n-1))}

Crossrefs

Cf. A144681, A144683, A144684.

Sequence in context: A276965 A217567 A048144 * A293146 A321189 A301387

Adjacent sequences: A144679 A144680 A144681 * A144683 A144684 A144685

Keyword

nonn

Author

Paul D. Hanna, Sep 19 2008

Status

approved