OFFSET
0,4
COMMENTS
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S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 14th equation.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
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]
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
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 21 2008
STATUS
approved