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A255365
Expansion of phi(-x^3) * phi(-x^48) / chi(-x^16) in powers of x where phi(), chi() are Ramanujan theta functions.
1
1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 3, 0, 0, -2
OFFSET
0,4
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 q^(-2/3) * eta(q^3)^2 * eta(q^32) * eta(q^48)^2 / (eta(q^6) * eta(q^16) * eta(q^96)) in powers of q.
Euler transform of a period 96 sequence.
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.
a(3*n) = A002448(n). a(3*n + 32) = A227395(n).
EXAMPLE
G.f. = 1 - 2*x^3 + 2*x^12 + x^16 - 2*x^19 - 2*x^27 + 2*x^28 + x^32 + ...
G.f. = q^2 - 2*q^11 + 2*q^38 + q^50 - 2*q^59 - 2*q^83 + 2*q^86 + q^98 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^16, x^16] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 3, 0, x^48], {x, 0, n}];
a[n_]:=SeriesCoefficient[EllipticTheta[3, 0, -x^3]*EllipticTheta[3, 0, -x^48 ]/QPochhammer[x^16, x^32], {x, 0, n}]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, March 14 2018 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^32 + A) * eta(x^48 + A)^2 / (eta(x^6 + A) * eta(x^16 + A) * eta(x^96 + A)), n))};
(PARI) {a(n) = my(A, p, e); if( n<0 || n%4==1, 0, n = 3*n+2; A = factor(n); - prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, -(e==1), if( p==3, 0, if( p%8 < 4, e+1, 1-e%2))))))};
CROSSREFS
Sequence in context: A084863 A368072 A233441 * A256505 A337196 A073346
KEYWORD
sign
AUTHOR
Michael Somos, Feb 21 2015
STATUS
approved