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
Keywords
Examples
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 + ...
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- S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 14th equation.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
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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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))};
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]
Comments