OFFSET
1,2
COMMENTS
Compare to identity: 1 + 2*Sum_{n>=0} x^(n^2) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n)^2 / (1 + x^(2*n))^2.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..260
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) B(x) = Product_{k>=1} (1 - x^(2*k)) * (1 + x^k - 2*A(x))^2 / (1 + x^(2*k) - 2*A(x))^2.
(2) B(x) = 1 - 2*Sum_{n>=1} (x^n - 2*A(x))^n.
(3) B(x) = 3 - 2*Sum_{n>=0} x^(n^2) / (1 + 2*x^n*A(x))^(n+1).
a(n) ~ c * d^n / n^(3/2), where d = 9.19219694975902875836550... and c = 0.0708465407554627020767... - Vaclav Kotesovec, Feb 19 2024
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 12*x^3 + 67*x^4 + 440*x^5 + 3080*x^6 + 22339*x^7 + 166958*x^8 + 1279405*x^9 + 10001248*x^10 + 79431437*x^11 + 639092704*x^12 + ...
By definition, A = A(x) allows for the following expressions to equal
B(x) = 1 - 2*(x - 2*A) - 2*(x^2 - 2*A)^2 - 2*(x^3 - 2*A)^3 - 2*(x^4 - 2*A)^4 - 2*(x^5 - 2*A)^5 + ...
B(x) = (1 - x^2)*(1 + x - 2*A)^2/(1 + x^2 - 2*A)^2 * (1 - x^4)*(1 + x^2 - 2*A)^2/(1 + x^4 - 2*A)^2 * (1 - x^6)*(1 + x^3 - 2*A)^2/(1 + x^6 - 2*A)^2 * (1 - x^8)*(1 + x^4 - 2*A)^2/(1 + x^8 - 2*A)^2 * ...
where B(x) begins
B(x) = 1 + 2*x + 4*x^2 + 24*x^3 + 138*x^4 + 872*x^5 + 5976*x^6 + 43104*x^7 + 321860*x^8 + 2464986*x^9 + 19256068*x^10 + 152848436*x^11 + 1229294528*x^12 + ...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( (1 - 2*sum(m=1, #A, (x^m - 2*Ser(A))^m ) - prod(m=1, #A, (1 - x^(2*m))*(1 + x^m - 2*Ser(A))^2/(1 + x^(2*m) - 2*Ser(A))^2))/4, #A-1)); H=A; A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 16 2024
STATUS
approved