A230534 Expansion of phi(x) / phi(x^2) * f(-x, -x^7) / f(-x^3, -x^5) in powers of x where phi(), f() are Ramanujan theta functions.

1, 1, -4, -1, 11, -1, -20, 2, 34, 1, -64, -2, 113, -2, -180, 2, 284, 4, -452, -4, 698, -5, -1044, 6, 1545, 6, -2276, -7, 3308, -9, -4724, 10, 6680, 13, -9408, -15, 13130, -17, -18112, 20, 24821, 21, -33864, -25, 45882, -28, -61696, 32, 82514, 39, -109892, -43

Offset: 0

Author

Michael Somos, Oct 22 2013

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x - 4*x^2 - x^3 + 11*x^4 - x^5 - 20*x^6 + 2*x^7 + 34*x^8 + ...
G.f. = q + q^3 - 4*q^5 - q^7 + 11*q^9 - q^11 - 20*q^13 + 2*q^15 + ...
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..10000 (first 2501 terms from G. C. Greubel)
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A080015, A092869, A226559.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 2 q^(-1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q]^2 (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]), {q, 0, n}];

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^6) * sum( k=1, sqrtint(n+1), x^k^2 - x^(2*k^2), x * A) / x, n))};

Formula

Expansion of (phi(x) / psi(x^2)^2) * (phi(x) - phi(x^2)) / (2*x) in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of q^(-1) * phi(q^2) * (phi(q^2) - phi(q^4)) / (phi(q^4) * (phi(q) - phi(q^4))) in powers of q^2 where phi() is a Ramanujan theta function.

Euler transform of period 8 sequence [1, -5, 3, 2, 3, -5, 1, 0, ...].

a(2*n + 1) = A226559(n).

Convolution of A080015 and A092869.