OFFSET
0,2
COMMENTS
REFERENCES
S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 147, Ed. G. H. Hardy et al., AMS Chelsea 2000.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. Josuat-Verges and J. S. Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, p. 28, equation (61), arXiv:1101.5608, 2011
FORMULA
Expansion of psi(x^2) * f(-x)^2 = phi(-x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -2, -1, -2, -3, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (u*w * (u + 2*w) * (u + 4*w))^2 - v^6 * (u^2 + 4*u*w + 8*w^2).
G.f.: Sum_{k} (3*k + 1) * x^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^(4*k)).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(4*n + 1) = -2 * a(n). 2 * a(n) = A113277(4*n + 1) = - A114855(4*n + 1).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 11 2015
EXAMPLE
G.f. = 1 - 2*x + 4*x^5 - 5*x^8 + 7*x^16 - 8*x^21 + 10*x^33 - 11*x^40 + ...
G.f. = q - 2*q^4 + 4*q^16 - 5*q^25 + 7*q^49 - 8*q^64 + 10*q^100 - 11*q^121 +...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := With[{m = Sqrt[3 n + 1]}, If[ IntegerQ[ m], -m (-1)^Mod[ m, 3], 0]]; (* Michael Somos, Mar 11 2015 *)
PROG
(PARI) {a(n) = if( issquare( 3*n + 1, &n), n * -(-1)^(n%3), 0)};
(PARI) {a(n) = local(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( e%2, 0, (-(-1)^(p%3) * p)^(e/2) )))) };
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^2 / eta(x^2 + A), n))};
(Magma) A := Basis( CuspForms( Gamma1(36), 3/2), 300); A[1] - 2*A[4]; /* Michael Somos, Mar 11 2015 */
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 01 2006
STATUS
approved