A209942 Expansion of (psi(-x) * phi(x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, 14, 81, 238, 322, 0, -429, -82, 0, -2162, -3038, 1134, 2401, -2482, 0, 6958, 3332, 0, 1442, 0, 6561, 4508, -9758, 0, -1918, -18802, 0, -9362, -24638, 19278, 14641, -14756, 0, 0, 6562, 0, -1148, 33998, 26082, 20398, 0, 0, 28083, -49042, 0, 64078, -30268, 0

Offset

0,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 60 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/4) * ( eta(q^2)^19 / (eta(q) * eta(q^4) )^7 )^2 in powers of q.

Euler transform of period 4 sequence [ 14, -24, 14, -10, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32768 (t/i)^5 f(t) where q = exp(2 Pi i t).

a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.

a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n). Convolution of A000143 and A134343.

Convolution square of A258771. - Michael Somos, Jun 09 2015

Example

G.f. = 1 + 14*x + 81*x^2 + 238*x^3 + 322*x^4 - 429*x^6 - 82*x^7 - 2162*x^9 + ...
G.f. = q + 14*q^5 + 81*q^9 + 238*q^13 + 322*q^17 - 429*q^25 - 82*q^29 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^19 / (QPochhammer[ x] QPochhammer[ x^4])^7)^2, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7 )^2, n))};

Crossrefs

Cf. A000143, A134343, A258771.

Sequence in context: A239421 A309338 A215472 * A215700 A199912 A082971

Adjacent sequences: A209939 A209940 A209941 * A209943 A209944 A209945

Keyword

sign

Author

Michael Somos, Mar 16 2012

Status

approved