login
A246811
Expansion of phi(x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
3
1, 4, 4, 0, 5, 12, 4, 0, 8, 12, 8, 0, 5, 16, 12, 0, 8, 24, 4, 0, 16, 12, 12, 0, 9, 24, 12, 0, 8, 36, 12, 0, 16, 12, 16, 0, 8, 28, 16, 0, 17, 36, 8, 0, 24, 24, 8, 0, 8, 36, 28, 0, 16, 36, 12, 0, 16, 24, 20, 0, 13, 24, 24, 0, 24, 60, 8, 0, 16, 36, 16, 0, 16, 28
OFFSET
0,2
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 psi(x)^4 / phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^2)^10 * eta(q^8)^2 / (eta(q)^4 * eta(q^4)^5) in powers of q.
Euler transform of period 8 sequence [4, -6, 4, -1, 4, -6, 4, -3, ...].
2 * a(n) = A033717(4*n + 2). a(2*n) = A045834(n). a(4*n) = A213022(n). a(4*n + 1) = 4 * A008443(n). a(4*n + 2) = 4 * A045831(n). a(4*n + 3) = 0.
EXAMPLE
G.f. = 1 + 4*x + 4*x^2 + 5*x^4 + 12*x^5 + 4*x^6 + 8*x^8 + 12*x^9 + ...
G.f. = q + 4*q^3 + 4*q^5 + 5*q^9 + 12*q^11 + 4*q^13 + 8*q^17 + 12*q^19 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, 0, x^2] / (2 x^(1/2)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^4 / (16 x^(1/2) EllipticTheta[ 3, 0, x^2]), {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^8 + A)^2 / (eta(x + A)^4 * eta(x^4 + A)^5), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 03 2014
STATUS
approved