login
A280384
Expansion of f(x)^3 * f(-x^2) * chi(x^3)^3 in powers of x where chi(), f() are Ramanujan theta functions.
3
1, 3, -1, -5, 8, -1, -28, 11, 10, -41, 41, 26, -53, 84, 21, -101, 76, 3, -129, 99, 14, -190, 187, 59, -299, 263, 62, -336, 340, 27, -459, 370, 111, -645, 518, 228, -774, 806, 179, -973, 882, 147, -1233, 955, 291, -1565, 1325, 395, -1883, 1767, 338, -2318, 1994
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Amanda Clemm, Modular Forms and Weierstrass Mock Modular Forms, Mathematics, volume 4, issue 1, (2016)
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q * eta(q^12)^10 * eta(q^36)^6 / (eta(q^6)^3 * eta(q^18)^3 * eta(q^24)^3 * eta(q^72)^3) in powers of q^6.
Euler transform of period 12 sequence [3, -7, 6, -4, 3, -10, 3, -4, 6, -7, 3, -4, ...].
a(n) = (-1)^n * A280328(n).
a(5*n + 1) / a(1) == A187076(n) (mod 5). a(125*n + 21) / a(21) == A187076(n) (mod 25).
EXAMPLE
G.f. = 1 + 3*x - x^2 - 5*x^3 + 8*x^4 - x^5 - 28*x^6 + 11*x^7 + 10*x^8 + ...
G.f. = q^-1 + 3*q^5 - q^11 - 5*q^17 + 8*q^23 - q^29 - 28*q^35 + 11*q^41 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 QPochhammer[ x^2] QPochhammer[ -x^3, x^6]^3, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 * eta(x^6 + A)^6 / (eta(x + A)^3 * eta(x^3 + A)^3 * eta(x^4 + A)^3 * eta(x^12 + A)^3), n))};
CROSSREFS
Sequence in context: A209831 A284367 A280328 * A124420 A176105 A094353
KEYWORD
sign
AUTHOR
Michael Somos, Jan 01 2017
STATUS
approved