A143063 Expansion of the product of a false theta function and a Ramanujan theta function in powers of x.

1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 2, 4, 4, 2, 4, 6, 4, 4, 4, 8, 8, 6, 8, 12, 10, 10, 12, 16, 16, 14, 18, 22, 22, 20, 24, 30, 32, 30, 36, 42, 42, 42, 48, 56, 60, 58, 66, 76, 78, 80, 88, 102, 106, 108, 120, 134, 140, 144, 158, 178, 186, 192, 210, 232, 242, 252, 272, 300

Offset

0,4

Comments

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

References

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 235, Entry 9.4.8

S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 14th equation.

Links

Table of n, a(n) for n=0..71.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: (1 - x + x^2 - x^5 + x^7 - x^12 + x^15 - ...) * (1 + x) * (1 + x^3) * (1 + x^5) * (1 + x^7) * ... [Ramanujan]

G.f.: 1 + 2 * x^3 / (1 - x^4) + 2 * x^8 / ((1 - x^2) * (1 - x^8)) + 2 * x^15 / ((1 - x^2) * (1 - x^4) * (1 - x^12)) + 2 * x^24 / ((1 - x^2) * (1 - x^4) * (1 - x^6) * (1 - x^16)) + ... [Ramanujan]

a(n) = 2 * A027348(n) unless n=0. Convolution of A143062 and A000700.

Example

G.f. = 1 + 2*x^3 + 2*x^7 + 2*x^8 + 2*x^10 + 2*x^11 + 2*x^12 + 2*x^14 + 4*x^15 + ...

Prog

(PARI) {a(n) = my(A, m); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=0, n, if( issquare( 24*k + 1, &m), (-1)^(m \ 3) * x^k ), A) / eta(x + A) * eta(x^2 + A)^2 / eta(x^4 + A), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, sqrtint(n+1) - 1, x^(k^2 + 2*k) / (1 - x^(4*k)) / prod(j=1, k-1, 1 - x^(2*j), 1 + O(x^(n + 1 - k^2 - 2*k)))), n))};

Crossrefs

Cf. A000700, A027348, A143062.

Sequence in context: A369309 A216228 A291957 * A210703 A340455 A275964

Adjacent sequences: A143060 A143061 A143062 * A143064 A143065 A143066

Keyword

nonn

Author

Michael Somos, Jul 21 2008

Status

approved