A170772 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.

4, 0, 16, 0, 16, 0, 0, 0, 32, 16, 64, 48, 0, 32, 16, 0, 96, 64, 48, 64, 96, 0, 80, 48, 0, 64, 128, 64, 16, 80, 0, 96, 144, 0, 128, 0, 112, 96, 128, 0, 224, 160, 0, 128, 240, 96, 208, 160, 0, 0, 304, 0, 256, 112, 192, 224, 32, 0, 240, 128, 0, 192, 224, 16, 336

Offset

0,1

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 right side of (27.1).

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

Example

G.f. = 4 + 16*x^2 + 16*x^4 + 32*x^8 + 16*x^9 + 64*x^10 + 48*x^11 + ...
G.f. = 4 + 16*q^4 + 16*q^8 + 32*q^16 + 16*q^18 + 64*q^20 + 48*q^22 + 32*q^26 + 16*q^28 + ...

Mathematica

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

Crossrefs

Cf. A170770, A170771, A170773.

Sequence in context: A060052 A059065 A170771 * A079986 A134746 A210067

Adjacent sequences: A170769 A170770 A170771 * A170773 A170774 A170775

Keyword

nonn

Author

Michael Somos, Dec 10 2009

Status

approved