login
A256269
Expansion of psi(-q) * chi(q^3) * phi(q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
6
1, -1, 0, 0, -1, 0, 0, 0, 0, 4, -2, 0, 0, -2, 0, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, -2, 0, 0, -2, 0, 0, 0, 0, 8, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, 4
OFFSET
0,10
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^18)^5 / (eta(q^2) * eta(q^3) * eta(q^12) * eta(q^9)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 2, 0, -1, -1, -1, 0, 0, -1, -1, -4, -1, -1, 0, 0, -1, -1, -1, 0, 2, -1, -1, -1, -1, -1, 0, 0, -1, -2, ...].
a(3*n + 1) = - A122865(n). a(6*n + 4) = - A122856(n). a(3*n + 2) = a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
EXAMPLE
G.f. = 1 - q - q^4 + 4*q^9 - 2*q^10 - 2*q^13 - q^16 + 4*q^18 - 3*q^25 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q^9], {q, 0, n}];
PROG
(PARI) {a(n) = if( n<1, n==0, (-1)^(n%3) * (n%3<2) * sumdiv(n, d, [ 0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^18 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};
(Magma) A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - A[2] - A[5] + 4*A[10] - 2*A[11] - 2*A[14] - A[17] + 4*A[19];
CROSSREFS
Sequence in context: A159257 A258997 A232833 * A256279 A363436 A277767
KEYWORD
sign
AUTHOR
Michael Somos, Jun 01 2015
STATUS
approved