OFFSET
0,2
COMMENTS
From Paul D. Hanna, Apr 24 2010: (Start)
SPECIAL VALUES:
. at x = exp(-Pi)*Pi^(-1/4)*gamma(3/4) = 0.039775896186087627425...,
. A(x) = theta_3(exp(-Pi)) = Pi^(1/4)/gamma(3/4) = 1.0864348112133080...
RADIUS OF CONVERGENCE r:
. at r = 0.241970723224463308846762732757915397312...,
. A(r) = 2.506628552782237708927560606516272396709...
where r and A(r) are given by:
. r = z/theta_3(z) and
. A(r) = theta_3(z)
such that z is the real root nearest the origin that satisfies:
. theta_3(z) - z*theta_3'(z) = 0, which has solution:
. z = 0.6065307237718078589943387177361885081872...
(End)
FORMULA
G.f.: A(x) = (1/x)*Series_Reversion(x/theta_3(x)).
G.f. satisfies: A(x/theta_3(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
G.f. satisfies: A(x) = Product_{n>=1} (1 - (-x)^n*A(x)^n) / (1 + (-x)^n*A(x)^n).
G.f. satisfies: A(x) = Product_{n>=1} (1 + (x*A(x))^(2*n-1))^2 * (1 - (x*A(x))^(2*n)).
G.f. satisfies: [x^n] A(x)^(1-n) = 2-2n if n>0 is square, zero otherwise.
a(n) = A066536(n)/(n+1) where A066536(n) equals the number of ways of writing n as the sum of n+1 squares.
Logarithmic derivative yields A066535, number of ways of writing n as the sum of n squares, for n>=1.
a(n) ~ c * d^n / n^(3/2), where d = 4.13273137623493996302796465513832835490078625705045019249993320055571... and c = 0.70710538549959357505200420443014251744770906948354300807129911827348... - Vaclav Kotesovec, Nov 16 2023
EXAMPLE
G.f.: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 18*x^4 + 52*x^5 + 184*x^6 +...
A(x/theta_3(x)) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 +...
log(A(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 24*x^4/4 + 112*x^5/5 +...+ A066535(n)*x^n/n +...
MATHEMATICA
CoefficientList[1/x*InverseSeries[Series[x/EllipticTheta[3, 0, x], {x, 0, 25}], x], x] (* Vaclav Kotesovec, Nov 16 2023 *)
(* Calculation of constants {d, c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)
PROG
(PARI) a(n)=local(THETA3=1+2*sum(k=1, sqrtint(n), x^(k^2))+x*O(x^n)); polcoeff(THETA3^(n+1), n)/(n+1)
(PARI) a(n)=local(A=1+x); for(i=1, n, A=prod(k=1, n, (1-(-x)^k*A^k+x*O(x^n))/(1+(-x)^k*A^k+x*O(x^n)) )); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", "))
(PARI) a(n)=local(A=1+x); for(i=1, n, A=prod(k=1, n, (1+(x*A)^(2*k-1)+x*O(x^n))^2*(1-(x*A)^(2*k)+x*O(x^n)) )); polcoeff(A, n)
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jul 12 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 26 2009
STATUS
approved