|
|
FORMULA
|
G.f.: Sum_{n=-oo..+oo} 2^(n*(n+1)/2) * x^(n^2) = Sum_{n>=0} a(n) * x^(n^2).
G.f.: Product_{n>=1} (1 - 2^n*x^(2*n)) * (1 + 2^n*x^(2*n-1)) * (1 + 2^(n-1)*x^(2*n-1)) = Sum_{n>=0} a(n) * x^(n^2), by the Jacobi triple product identity.
a(2*n+1) = 0 (mod 3), a(4*n+2) = 1 (mod 3), a(4*n+4) = 2 (mod 3), for n >= 0 (conjecture).
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 3*x + 10*x^4 + 72*x^9 + 1088*x^16 + 33792*x^25 + 2129920*x^36 + 270532608*x^49 + 68987912192*x^64 + 35253091565568*x^81 + 36063981391052800*x^100 + ... + a(n)*x^(n^2) + ...
which can be generated by the Jacobi Triple Product:
A(x) = (1 - 2*x^2)*(1 + 2*x)*(1 + x) * (1 - 2^2*x^4)*(1 + 2^2*x^3)*(1 + 2*x^3) * (1 - 2^3*x^6)*(1 + 2^3*x^5)*(1 + 2^2*x^5) * (1 - 2^4*x^8)*(1 + 2^4*x^7)*(1 + 2^3*x^7) * ... * (1 - 2^n*x^(2*n))*(1 + 2^n*x^(2*n-1))*(1 + 2^(n-1)*x^(2*n-1)) * ...
|
|
|
PROG
|
(PARI) {a(n) = if(n==0, 1, 2^(n*(n-1)/2) + 2^(n*(n+1)/2))}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* As Coefficients in a Jacobi Theta Function: */
{a(n) = polcoeff( sum(m=-n, n, 2^(m*(m+1)/2)*x^(m^2) +x*O(x^(n^2))), n^2)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* By the Jacobi Triple Product identity: */
{a(n) = polcoeff( prod(m=1, n^2\2+1, (1 - 2^m*x^(2*m)) * (1 + 2^m*x^(2*m-1)) * (1 + 2^(m-1)*x^(2*m-1)) +x*O(x^(n^2))), n^2)}
for(n=0, 15, print1(a(n), ", "))
|
