OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Johann Cigler, Some Pascal-like triangles, 2018.
Ken Ono and Christopher Skinner, Fourier Coefficients of Half-Integral Weight Modular Forms Modulo l, Ann. Math., 147 (1998), 453-470.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: Product_{k>0} (1 - x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k)). - Michael Somos, Sep 21 2005
Euler transform of period 8 sequence [2, -1, 2, -5, 2, -1, 2, -4, ...]. - Michael Somos, Sep 21 2005
Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^4) / (eta(q)^2 * eta(q^8)) in powers of q. - Michael Somos, Sep 21 2005
Expansion of phi(x) * f(x^2)^2 * f(-x^8) = psi(x)^2 * f(x^2) * f(-x^4) = psi(x)^2 * psi(-x^2) * phi(x^2) = psi(x^2)^2 * phi(x) * phi(-x^4) = psi(x)^2 * psi(x^2) * phi(-x^4) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Jul 07 2014
a(31*n + 15) = 0 unless n == 15 (mod 31). a(961*n + 480) = -31 * a(n). - Michael Somos, Jul 07 2014
EXAMPLE
G.f. = 1 + 2*x + 2*x^2 + 4*x^3 + x^4 - 2*x^5 + 2*x^6 - 4*x^7 - 2*x^8 + ...
G.f. = q + 2*q^3 + 2*q^5 + 4*q^7 + q^9 - 2*q^11 + 2*q^13 - 4*q^15 - 2*q^17 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^2 EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^4] / (4 x^(1/2)), {x, 0, n}];
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^4)/(QP[q]^2*QP[q^8]) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^4 / (eta(x + A)^2 * eta(x^8 + A)), n))}; /* Michael Somos, Sep 21 2005 */
CROSSREFS
KEYWORD
sign
AUTHOR
STATUS
approved