OFFSET
0,3
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(2*n-1) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(2*n^2-n) / (1 + (1+x)^(2*n)*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 60*x^3 + 1349*x^4 + 40210*x^5 + 1470027*x^6 + 63225750*x^7 + 3116555468*x^8 + 172936040306*x^9 + 10661699020596*x^10 + ...
such that
1 = 1 + ((1+x) - A(x)) + ((1+x)^3 - A(x))^2 + ((1+x)^5 - A(x))^3 + ((1+x)^7 - A(x))^4 + ((1+x)^9 - A(x))^5 + ((1+x)^11 - A(x))^6 + ((1+x)^13 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)/(1 + (1+x)^2*A(x))^2 + (1+x)^6/(1 + (1+x)^4*A(x))^3 + (1+x)^15/(1 + (1+x)^6*A(x))^4 + (1+x)^28/(1 + (1+x)^8*A(x))^5 + (1+x)^45/(1 + (1+x)^10*A(x))^6 + (1+x)^66/(1 + (1+x)^6*A(x))^7 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(2*m-1) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 12 2019
STATUS
approved