A209942 - OEIS

%I #19 Mar 12 2021 22:24:46

%S 1,14,81,238,322,0,-429,-82,0,-2162,-3038,1134,2401,-2482,0,6958,3332,

%T 0,1442,0,6561,4508,-9758,0,-1918,-18802,0,-9362,-24638,19278,14641,

%U -14756,0,0,6562,0,-1148,33998,26082,20398,0,0,28083,-49042,0,64078,-30268,0

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

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

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

%H G. C. Greubel, <a href="/A209942/b209942.txt">Table of n, a(n) for n = 0..1000</a>

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

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

%F 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).

%F 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.

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

%F Convolution square of A258771. - _Michael Somos_, Jun 09 2015

%e G.f. = 1 + 14*x + 81*x^2 + 238*x^3 + 322*x^4 - 429*x^6 - 82*x^7 - 2162*x^9 + ...

%e G.f. = q + 14*q^5 + 81*q^9 + 238*q^13 + 322*q^17 - 429*q^25 - 82*q^29 + ...

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

%o (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))};

%Y Cf. A000143, A134343, A258771.

%K sign

%O 0,2

%A _Michael Somos_, Mar 16 2012