A228447 Expansion of q * (psi(q^3) * psi(q^6)) / (psi(q) * phi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.

1, -3, 7, -15, 30, -57, 104, -183, 313, -522, 852, -1365, 2150, -3336, 5106, -7719, 11538, -17067, 25004, -36306, 52280, -74700, 105960, -149277, 208951, -290706, 402127, -553224, 757158, -1031166, 1397744, -1886151, 2534316, -3391254, 4520112, -6002007

Offset

1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q * (psi(q^3)^3 / psi(q)) / (phi(q) * phi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.

Expansion of eta(q)^3 * eta(x^4)^2 * eta(x^6) * eta(x^12)^2 / (eta(x^2)^7 * eta(x^3)) in powers of q.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 * (1 - 3*v) - v * (1 - 4*v) * (1 - 3*u)^2.

a(n) = -(-1)^n * A187100(n). a(2*n) = -3 * A128638(n).

Convolution inverse is A187145. Convolution with A033716 is A093829.

Example

G.f. = q - 3*q^2 + 7*q^3 - 15*q^4 + 30*q^5 - 57*q^6 + 104*q^7 - 183*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^(3/2)]^3 / (EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3]), {q, 0, n}]

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x^2 + A)^7 * eta(x^3 + A)), n))}

Crossrefs

Cf. A033716, A093829, A128638, A187100, A187145.

Sequence in context: A290865 A055795 A058695 * A187100 A209816 A182726

Adjacent sequences: A228444 A228445 A228446 * A228448 A228449 A228450

Keyword

sign

Author

Michael Somos, Oct 26 2013

Status

approved