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..261
FORMULA
G.f. A(x) = Sum_{n>=0} 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} Product_{k=1..n} (x^(2*k-1) - 2*A(x)).
(3) B(x) = 3 - 2*Sum_{n>=0} x^(n^2) / Product_{k=0..n} (1 + 2*x^(2*k)*A(x)).
(4) B(x) = 1 - 2*(x - 2*A(x))/(1 + F(1)), where F(n) = -(x^(2*n+1) - 2*A(x))/(1 + (x^(2*n+1) - 2*A(x)) + F(n+1)), a continued fraction.
a(n) ~ c * d^n / n^(3/2), where d = 8.5161154068976325332006245928737661345... and c = 0.06900087714889532378308608613059862... - Vaclav Kotesovec, Feb 27 2024
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 9*x^3 + 49*x^4 + 295*x^5 + 1893*x^6 + 12697*x^7 + 87985*x^8 + 625050*x^9 + 4527978*x^10 + 33321906*x^11 + 248416364*x^12 + ...
By definition, A = A(x) allows for the following expressions to equal
B(x) = 1 - 2*(x - 2*A) - 2*(x - 2*A)*(x^3 - 2*A) - 2*(x - 2*A)*(x^3 - 2*A)*(x^5 - 2*A) - 2*(x - 2*A)*(x^3 - 2*A)*(x^5 - 2*A)*(x^7 - 2*A) - 2*(x - 2*A)*(x^3 - 2*A)*(x^5 - 2*A)*(x^7 - 2*A)*(x^9 - 2*A) + ...
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 + 20*x^3 + 106*x^4 + 628*x^5 + 3996*x^6 + 26676*x^7 + 184336*x^8 + 1307010*x^9 + 9454904*x^10 + 69505040*x^11 + 517724884*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, prod(k=1, m, x^(2*k-1) - 2*Ser(A)) ) - 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