A170770 Expansion of ( phi(q) * phi(q^63) + phi(-q) * phi(-q^63) + 4 * q^16 * psi(q^2) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions.

1, 0, 2, 0, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 8, 0, 4, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2

Offset

0,3

Comments

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

References

Bruce C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, 1984; see Entry 27, pp. 170-171. This is the square root of the right side of (27.1), divided by 2.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Michael Somos, Notes on Entry 27 of Chapter 25 of Ramanujan's Notebooks, Part IV

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: ( phi(q^7) * phi(q^9) + phi(-q^7) * phi(-q^9) + 4 * q^4 * psi(q^14) * psi(q^18) ) / 2 in powers of q^2 where phi(), psi() are Ramanujan theta functions. - Michael Somos, May 24 2018

Example

G.f. = 1 + 2*x^2 + 4*x^8 + 2*x^9 + 2*x^11 + 2*x^14 + 4*x^18 + 2*x^23 + ...
G.f. = 1 + 2*q^4 + 4*q^16 + 2*q^18 + 2*q^22 + 2*q^28 + 4*q^36 + 2*q^46 + 2*q^58 + ...

Mathematica

QP = QPochhammer; p[q_] := EllipticTheta[3, 0, q]; u[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, Sqrt[q]]; a[n_] := SeriesCoefficient[(1/2)*(p[q]*p[q^63] + p[-q]*p[-q^63] + 4*q^16*u[q^2]*u[q^(126)]), {q, 0, n}]; Table[a[n], {n, 0, 100}][[ ;; ;; 2]] (* G. C. Greubel, Dec 05 2017 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^7] EllipticTheta[ 3, 0, q^9] + 1/2 EllipticTheta[ 2, 0, q^7] EllipticTheta[ 2, 0, q^9], {q, 0, 2 n}, Assumptions -> q>0]; (* Michael Somos, May 24 2018 *)

Crossrefs

Cf. A170771, A170772, A170773.

Sequence in context: A005871 A005888 A213902 * A107499 A123298 A189877

Adjacent sequences: A170767 A170768 A170769 * A170771 A170772 A170773

Keyword

nonn

Author

Michael Somos, Dec 10 2009

Status

approved