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A208845
Expansion of f(x)^2 in powers of x where f() is a Ramanujan theta function.
2
1, 2, -1, -2, 1, -2, -2, 0, -2, 2, 1, 0, 0, -2, 3, 2, 2, 0, 0, 2, -2, 0, 0, 2, -1, 0, 2, -2, -2, -2, 1, -2, 0, -2, -2, 2, 2, 0, -2, 0, -4, 0, 0, 0, 1, 2, 0, 0, 2, 0, 2, -2, 1, 2, 0, -2, 2, 0, 0, 2, 0, 2, 0, 2, 2, 0, -4, 0, 0, 2, -1, -2, 0, -2, 0, 0, 0, 2, 2
OFFSET
0,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 73 of the 74 eta-quotients listed in Table I of Martin (1996).
LINKS
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(-1/12) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^2 in powers of q.
Euler transform of period 4 sequence [ 2, -4, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = (24)^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
a(n) = b(12*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 5 (mod 12), b(p^e) = (e + 1) * (-1)^(e * ((p%24>1) + x)) if p == 1 (mod 12) and p = x^2 + 9 * y^2.
a(n) = (-1)^n * A002107(n). a(25*n + 2) = -a(n).
Convolution cube is A209941. - Michael Somos, Jun 09 2015
EXAMPLE
G.f. = 1 + 2*x - x^2 - 2*x^3 + x^4 - 2*x^5 - 2*x^6 - 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q + 2*q^13 - q^25 - 2*q^37 + q^49 - 2*q^61 - 2*q^73 - 2*q^97 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 , {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
PROG
(PARI) {a(n) = my(A, p, e, m); if( n<0, 0, n = 12*n + 1; A=factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 0, p%12>1, if( e%2, 0, (-1)^( (p%12==5) * e/2)), for( i=1, sqrtint(p\9), if( issquare( p - 9*i^2), m=i; break)); (e+1) * (-1)^(e * ( (p%24>1) + m )))))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A) * eta(x^4 + A)))^2, n))};
CROSSREFS
Sequence in context: A335665 A369239 A002107 * A232506 A133099 A006571
KEYWORD
sign
AUTHOR
Michael Somos, Mar 03 2012
STATUS
approved