OFFSET
0,4
COMMENTS
Compare the e.g.f. to the identities:
(1) G(x) = exp( Integral G(x)^t dx ) when G(x) = 1/(1-t*x)^(1/t).
(2) G(x) = 1 + Integral G(x)^(1+t) dx when G(x) = 1/(1-t*x)^(1/t).
All terms appear to be odd.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
E.g.f.: A(x) = 1 + Integral A(x)^(1+x) dx.
E.g.f.: A(x) = 1/A(-x).
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 9*x^4/4! + 41*x^5/5! + 201*x^6/6! + 1251*x^7/7! + 8433*x^8/8! + 66929*x^9/9! + 572081*x^10/10! + 5531491*x^11/11! + 57181881*x^12/12! +...
where A(x) = exp( Integral A(x)^x dx ),
also, A(x) = 1 + Integral A(x)^(1+x) dx.
RELATED SERIES.
The e.g.f. satisfies: A(x)' = A(x)^(1+x), which begins:
A(x)^(1+x) = 1 + x + 3*x^2/2! + 9*x^3/3! + 41*x^4/4! + 201*x^5/5! + 1251*x^6/6! + 8433*x^7/7! + + 66929*x^8/8! + 572081*x^9/9! + 5531491*x^10/10! +...
The series A(x)^x = A(x)'/A(x) is an even function that begins:
A(x)^x = 1 + 2*x^2/2! + 20*x^4/4! + 480*x^6/6! + 21200*x^8/8! + 1495040*x^10/10! + 154090560*x^12/12! + 21851648000*x^14/14! +...+ A274738(n)*x^(2*n)/(2*n)! +...
Compare to the logarithm of A(x), an odd function which begins:
log(A(x)) = x + 2*x^3/3! + 20*x^5/5! + 480*x^7/7! + 21200*x^9/9! + 1495040*x^11/11! + 154090560*x^13/13! +...+ A274738(n)*x^(2*n+1)/(2*n+1)! +...
thus 1/A(-x) = A(x).
PROG
(PARI) {a(n) = my(A=1); for(i=0, n, A = 1 + intformal( A^(1+x) +x*O(x^n) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=0, n, A = exp( intformal( A^x +x*O(x^n) ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 05 2016
STATUS
approved