OFFSET
1,3
FORMULA
G.f. B(x) and A(x) satisfy the differential series:
(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * B(x)^n / n!.
(2) B(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * A(x)^n / n!.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * B(x)^n / (n!*x) ).
(4) B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * A(x)^n / n! ).
EXAMPLE
G.f.: B(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 615*x^6 + 5038*x^7 + 45265*x^8 + 437012*x^9 + 4472197*x^10 + 48056889*x^11 +...
such that A(x - A(x)*B(x)) = x and B(x - x*A(x)) = x where
A(x) = x + x^2 + 4*x^3 + 23*x^4 + 160*x^5 + 1260*x^6 + 10861*x^7 + 100474*x^8 + 984944*x^9 + 10142888*x^10 + 109039530*x^11 +...
PROG
(PARI) {a(n) = my(A=x, B=x); for(i=1, n, A = serreverse(x - A*B +x*O(x^n)); B=serreverse(x - x*A); ); polcoeff(B, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 03 2015
STATUS
approved