OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} x^n * (1-x)^n / ( (1-x)^n + x*A(x) )^(n+1).
(2) 1 = Sum_{n>=0} x^n * ( 1 - A(x)*(1-x)^n )^n / (1-x)^(n^2).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 17*x^5 + 69*x^6 + 310*x^7 + 1530*x^8 + 8079*x^9 + 45325*x^10 + 268362*x^11 + 1667358*x^12 + ...
where
1 = 1 + x*(1/(1-x) - A(x)) + x^2*(1/(1-x)^2 - A(x))^2 + x^3*(1/(1-x)^3 - A(x))^3 + x^4*(1/(1-x)^4 - A(x))^4 + x^5*(1/(1-x)^5 - A(x))^5 + ...
Also,
1 = 1/(1 + x*A(x)) + x*(1-x)/((1-x) + x*A(x))^2 + x^2*(1-x)^2/((1-x)^2 + x*A(x))^3 + x^3*(1-x)^3/((1-x)^3 + x*A(x))^4 + x^4*(1-x)^4/((1-x)^4 + x*A(x))^5 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=1, #A, x^m*(1/(1-x)^m - Ser(A))^m), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 29 2018
STATUS
approved