OFFSET
0,2
COMMENTS
Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is square, else a(n) is divisible by 4 if n is nonsquare.
What is the radius of convergence of g.f. A(x)?
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..630
Wikipedia, Theta function.
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas, in which B(x) = Series_Reversion( x/A(x) ).
(1) A(x/A(x)) = 1 + 2*Sum_{n>=1} x^(n^2).
(2) A(x) = 1 + 2*Sum_{n>=1} B(x)^(n^2).
(3) A(x) = 1 + 2*Product_{n>=1} (1 - (-B(x))^n) / (1 + (-B(x))^n).
(4) A(x) = 1 + 2*Product_{n>=1} (1 + B(x)^(2*n-1))^2 * (1 - B(x)^(2*n)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^2 + 16*x^3 + 98*x^4 + 756*x^5 + 6848*x^6 + 70000*x^7 + 787056*x^8 + 9569826*x^9 + 124370360*x^10 + ...
where A(x/A(x)) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + ... + 2*x^(n^2) + ...
RELATED SERIES.
x/A(x) = x - 2*x^2 - 8*x^4 - 50*x^5 - 428*x^6 - 4152*x^7 - 44704*x^8 - 523244*x^9 - 6569450*x^10 - ...
B(x) = Series_Reversion( x/A(x) ) = x + 2*x^2 + 8*x^3 + 48*x^4 + 370*x^5 + 3368*x^6 + 34584*x^7 + 390112*x^8 + 4753856*x^9 + 61875338*x^10 + ...
where A(x) = 1 + 2*Sum_{n>=1} B(x)^(n^2).
SPECIAL VALUES.
A(exp(-2*Pi)) = 1.0037489402407331697131099797749462699662870856463...
with exp(-2*Pi) / A(exp(-2*Pi)) = 0.0018604679485490788842340854296021527548636828...
where A( exp(-2*Pi) / A(exp(-2*Pi)) ) = Pi^(1/4)/gamma(3/4) * sqrt(2 + sqrt(2))/2.
A(exp(-3*Pi)) = 1.0001614250932020887172209235064321667919549956244...
with exp(-3*Pi) / A(exp(-3*Pi)) = 0.0000806864927456929760953733590592766400756737...
where A( exp(-3*Pi) / A(exp(-3*Pi)) ) = Pi^(1/4)/gamma(3/4) * sqrt(1 + sqrt(3))/108^(1/8).
A(exp(-4*Pi)) = 1.0000069747333593234273049958805480741254454671485...
with exp(-4*Pi) / A(exp(-4*Pi)) = 0.0000034873180330955754899049884144807436643893...
where A( exp(-4*Pi) / A(exp(-4*Pi)) ) = Pi^(1/4)/gamma(3/4) * (2 + 8^(1/4))/4.
A(exp(-5*Pi)) = 1.0000003014035459221104165806302902976824246416705...
with exp(-5*Pi) / A(exp(-5*Pi)) = 0.0000001507016821169850945888085644257195104814...
where A( exp(-5*Pi) / A(exp(-5*Pi)) ) = Pi^(1/4)/gamma(3/4) * sqrt((2 + sqrt(5))/5).
PROG
(PARI) {a(n) = my(A=[1], Theta3 = 1 + 2*sum(m=1, sqrtint(n+1), x^(m^2) +x*O(x^n) ));
for(i=1, n, A=concat(A, 0);
A[#A] = polcoef( Theta3 - subst(Ser(A), x, x/Ser(A)), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Jan 08 2025
STATUS
approved