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A239051
Expansion of (f(-q^2, -q^3)^5 - 3 * q * f(-q, -q^4)^5) / f(-q)^3 in powers of q where f() is a Ramanujan theta function.
1
1, 0, 10, -10, 10, 0, 0, 10, 0, -10, 10, 0, 10, -10, 20, -10, 0, 10, -10, 0, 10, 0, 20, -10, 0, 0, 0, 0, 10, 0, 0, 0, 10, -20, 20, 10, 0, 10, 0, -20, 0, 0, 20, -10, 20, -10, 0, 10, -10, 10, 10, 0, 10, -10, 0, 0, 0, 0, 0, 0, 10, 0, 20, -10, 10, -10, 0, 10, 10
OFFSET
0,3
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
Moebius transform is period 5 sequence [ 0, 10, -10, 0, 0, ...].
G.f.: 1 + 10 * ( Sum_{k>=0} x^(5*k + 2) / (1 - x^(5*k + 2)) - x^(5*k + 3) / (1 - x^(5*k + 3)) ).
a(n) = A227216(n) - 3 * A229802(n).
a(5*n) = a(n). a(5*n + 1) = 0.
EXAMPLE
G.f. = 1 + 10*q^2 - 10*q^3 + 10*q^4 + 10*q^7 - 10*q^9 + 10*q^10 + 10*q^12 + ...
MATHEMATICA
a[ n_] := If[ n < 1, Boole[ n == 0], 10 Sum[ {0, 1, -1, 0, 0}[[ Mod[ d, 5, 1] ]] ], {d, Divisors @ n}]];
PROG
(PARI) {a(n) = if( n<1, n==0, 10 * sumdiv(n, d, (d%5==2) - (d%5==3)))};
(Sage) ModularForms( Gamma1(5), 1, prec=70).0;
(Magma) Basis( ModularForms( Gamma1(5), 1), 70) [1];
CROSSREFS
Sequence in context: A087028 A145279 A103708 * A131722 A072803 A163139
KEYWORD
sign
AUTHOR
Michael Somos, Jun 13 2014
STATUS
approved