A217567 E.g.f. satisfies: A(x) = ( Sum_{n>=0} x^n/n!^2 )^A(x) where A(x) = Sum_{n>=0} a(n)*x^n/n!^2.

1, 1, 5, 73, 2061, 97301, 6897203, 686934284, 91511132653, 15722347919797, 3385861914011775, 893404629519870524, 283510131741909375339, 106536362337513833330932, 46788887175103244923057374, 23747979495191419502491847783, 13795147423164719523469062474093

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n!^2 satisfies the following formulas.

(1) A(x) = ( Sum_{n>=0} x^n/n!^2 )^A(x).

(2) A(x) = log(A(x)) / ( Sum_{n>=0} A101981(n)*x^n/n!^2 ).

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2!^2 + 73*x^3/3!^2 + 2061*x^4/4!^2 + 97301*x^5/5!^2 +...
where
A(x) = (1 + x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 + x^5/5!^2 +...)^A(x).
Related expansions:
log(A(x)) = x + 3*x^2/2!^2 + 31*x^3/3!^2 + 679*x^4/4!^2 + 25581*x^5/5!^2 + 1474706*x^6/6!^2 + 120670201*x^7/7!^2 + 13298986863*x^8/8!^2 +...
log(A(x))/A(x) = log(1 + x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 +...);
log(A(x))/A(x) = x - x^2/2!^2 + 4*x^3/3!^2 - 33*x^4/4!^2 + 456*x^5/5!^2 - 9460*x^6/6!^2 + 274800*x^7/7!^2 +...+ A101981(n)*x^n/n!^2 +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/m!^2+x*O(x^n))^A); n!^2*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A101981, A179156.

Sequence in context: A197764 A126748 A276965 * A048144 A144682 A293146

Adjacent sequences: A217564 A217565 A217566 * A217568 A217569 A217570

Keyword

nonn

Author

Paul D. Hanna, Oct 07 2012

Status

approved