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A228746
Expansion of 8 * phi(q)^4 - 7 * phi(-q)^4 in powers of q where phi() is a Ramanujan theta function.
3
1, 120, 24, 480, 24, 720, 96, 960, 24, 1560, 144, 1440, 96, 1680, 192, 2880, 24, 2160, 312, 2400, 144, 3840, 288, 2880, 96, 3720, 336, 4800, 192, 3600, 576, 3840, 24, 5760, 432, 5760, 312, 4560, 480, 6720, 144, 5040, 768, 5280, 288, 9360, 576, 5760, 96, 6840
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution with A005875 is A004008.
LINKS
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 .
a(2*n) = A004011(n). a(2*n + 1) = 120 * A008438(n).
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 */
KEYWORD
nonn,easy
AUTHOR
Michael Somos, Sep 02 2013
STATUS
approved