OFFSET
0,2
COMMENTS
Compare: G(x) = (1 + Integral G(x) dx)^2 holds when G(x) = 1/(1 - x)^2.
Compare: G(x) = (1 + Integral G(x)^2 dx)^2 holds when G(x) = 1/(1 - 3*x)^(2/3), the e.g.f. of the triple factorials product_{k=0..n-1} (3*k+2).
Compare: G(x) = (1 + Integral G(x)^m dx)^2 holds when G(x) = 1/(1 - (2*m-1)*x)^(2/(2*m-1)) = Sum_{n>=0} x^n/n! * product_{k=0..n-1} ((2*m-1)*k + 2).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..397
FORMULA
E.g.f. A(x) satisfies the following relations.
(1) A(x) = (1 + Integral A(x) dx) * (1 + Integral A(x)^2 dx).
(2) A'(x) = A(x) * (1 + Integral A(x)^2 dx) + A(x)^2 * (1 + Integral A(x) dx).
(3) log(A(x)) = Integral [ A(x)/(1 + Integral A(x) dx) + A(x)^2/(1 + Integral A(x)^2 dx) ] dx.
(4a) log(1 + Integral A(x) dx) = Integral (1 + Integral A(x)^2 dx) dx.
(4b) log(1 + Integral A(x)^2 dx) = Integral A(x)*(1 + Integral A(x) dx) dx.
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 50*x^3/3! + 422*x^4/4! + 4480*x^5/5! + 57300*x^6/6! + 857364*x^7/7! + 14690244*x^8/8! + 283594200*x^9/9! + 6090223440*x^10/10! + ...
RELATED SERIES.
A(x)^2 = 1 + 4*x + 24*x^2/2! + 196*x^3/3! + 2028*x^4/4! + 25400*x^5/5! + 373400*x^6/6! + 6301408*x^7/7! + 120040416*x^8/8! + 2547619968*x^9/9! + ...
log(A(x)) = 2*x + 4*x^2/2! + 18*x^3/3! + 118*x^4/4! + 1028*x^5/5! + 11180*x^6/6! + 145784*x^7/7! + 2216600*x^8/8! + 38502688*x^9/9! + 752186400*x^10/10! + ...
such that
log(A(x)) = Integral [ (1 + Integral A(x)^2 dx) + A(x)*(1 + Integral A(x) dx) ] dx.
PROG
(PARI) {a(n) = my(A=1); for(i=1, n, A = (1 + intformal( A^1 )) * (1 + intformal( A^2 +x*O(x^n))) ); n!*polcoeff(H=A, n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 14 2019
STATUS
approved