OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
a(n) = 120 * b(n) with b() multiplicative where b(2^e) = 1/5 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228745.
G.f.: 8 * (Sum_{k in Z} x^k^2)^4 - 7 * (Sum_{k in Z} (-x)^k^2)^4 .
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4*Pi^2 = 39.478417... (A212002). - Amiram Eldar, Dec 29 2023
EXAMPLE
G.f. = 1 + 120*q + 24*q^2 + 480*q^3 + 24*q^4 + 720*q^5 + 96*q^6 + 960*q^7 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( 8 * A^4 - 7 * subst(A, x, -x)^4, n))};
(Magma) A := Basis( ModularForms( Gamma0(4), 2), 50); A[1] + 120*A[2]; /* Michael Somos, Aug 21 2014 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Sep 02 2013
STATUS
approved