OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) 1+x = Sum_{n>=0} ((1+x)^(n+1) - A(x))^n.
(2) 1+x = Sum_{n>=0} (1+x)^(n*(n+1)) / (1 + (1+x)^n*A(x))^(n+1).
a(n) ~ c * r^(2*n) * (1 + exp(1/r))^n * n^n / exp(n), where r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation (1 + exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1/r, and c = 0.9243009841585606... - Vaclav Kotesovec, Aug 06 2018
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 215*x^4 + 3221*x^5 + 60255*x^6 + 1328529*x^7 + 33525557*x^8 + 949932155*x^9 + 29820028655*x^10 + ...
such that
1+x = 1 + ((1+x)^2 - A(x)) + ((1+x)^3 - A(x))^2 + ((1+x)^4 - A(x))^3 + ((1+x)^5 - A(x))^4 + ((1+x)^6 - A(x))^5 + ((1+x)^7 - A(x))^6 + ((1+x)^8 - A(x))^7 + ...
Also,
1+x = 1/(1 + A(x)) + (1+x)^2/(1 + (1+x)*A(x))^2 + (1+x)^6/(1 + (1+x)^2*A(x))^3 + (1+x)^12/(1 + (1+x)^3*A(x))^4 + (1+x)^20/(1 + (1+x)^4*A(x))^5 + (1+x)^30/(1 + (1+x)^5*A(x))^6 + (1+x)^42/(1 + (1+x)^6*A(x))^7 + ...
PROG
(PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(m+1) - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 02 2018
STATUS
approved