A170773 Expansion of ( phi(q) * phi(q^7) * phi(q^9) * phi(q^63) + phi(-q) * phi(-q^7) * phi(-q^9) * phi(-q^63) + 4 * q^4 * f(-q^6)^2 * f(-q^42)^2 + 16 * q^20 * psi(q^2) * psi(q^14) * psi(q^18) * psi(q^126) ) / 2 in powers of q^2 where phi(), psi(), and f() are Ramanujan theta functions.

1, 0, 4, 0, 4, 0, 0, 0, 8, 4, 16, 12, 0, 8, 4, 0, 24, 16, 12, 16, 24, 0, 20, 12, 0, 16, 32, 16, 4, 20, 0, 24, 36, 0, 32, 0, 28, 24, 32, 0, 56, 40, 0, 32, 60, 24, 52, 40, 0, 0, 76, 0, 64, 28, 48, 56, 8, 0, 60, 32, 0, 48, 56, 4, 84, 48, 0, 48, 88, 0, 16, 36, 60, 40, 80, 0

Offset

0,3

Comments

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

This is the convolution square of A170770.

References

Bruce C. Berndt, Ramanujan's Notebooks, Part IV, Springer-Verlag, 1984; see Entry 27, pp. 170-171. This is the corrected left side of (27.1), divided by 4.

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

Example

G.f. = 1 + 4*x^2 + 4*x^4 + 8*x^8 + 4*x^9 + 16*x^10 + 12*x^11 + 8*x^13 + ...
G.f. = 1 + 4*q^4 + 4*q^8 + 8*q^16 + 4*q^18 + 16*q^20 + 12*q^22 + 8*q^26 + 4*q^28 + ...

Mathematica

f[q_] := QPochhammer[-q, -q]; p[q_] := EllipticTheta[3, 0, q]; u[q_] := (1/2)*q^(-1/8)*EllipticTheta[2, 0, Sqrt[q]]; A170773[n_] := SeriesCoefficient[(1/2)*(p[q]*p[q^7]*p[q^9]*p[q^63] + p[-q]*p[-q^7]*p[-q^9]*p[-q^63] + 4*q^4*f[-q^6]^2*f[-q^42]^2 + 16*q^20*u[q^2]*u[q^14]*u[q^18]*u[q^(126)]), {q, 0, n}]; Table[A170773[n], {n, 0, 100}][[ ;; ;; 2]] (* G. C. Greubel, Dec 03 2017 *)

Crossrefs

Cf. A170770-A170772.

Sequence in context: A066273 A028650 A028634 * A028618 A147986 A147988

Adjacent sequences: A170770 A170771 A170772 * A170774 A170775 A170776

Keyword

nonn

Author

Michael Somos, Dec 10 2009

Status

approved