A133089 - OEIS

%I #12 Mar 12 2021 22:24:44

%S 1,3,0,-5,0,0,-7,0,0,0,9,0,0,0,0,11,0,0,0,0,0,-13,0,0,0,0,0,0,-15,0,0,

%T 0,0,0,0,0,17,0,0,0,0,0,0,0,0,19,0,0,0,0,0,0,0,0,0,-21,0,0,0,0,0,0,0,

%U 0,0,0,-23,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0

%N Expansion of f(x)^3 in powers of x where f() is a Ramanujan theta function.

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

%D S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266.  MR0099904 (20 #6340)

%H Seiichi Manyama, <a href="/A133089/b133089.txt">Table of n, a(n) for n = 0..10000</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^(-1/8) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^3 in powers of q.

%F Euler transform of period 4 sequence [ 3, -6, 3, -3, ...].

%F a(n) = b(8*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 64 (t/i)^(3/2) f(t) where q = exp(2 Pi i t).

%F a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = 3 * a(n). a(25*n + 3) = -5 * a(n).

%F G.f.: Sum_{k>=0} (-1)^floor(k/2) * (2*k + 1) * x^(k*(k + 1))/2.

%F G.f.: ( Product_{k>0} (1 - x^k) * (1 + x^k)^2 / (1 + x^(2*k)) )^3.

%F a(n) = -(-1)^n * A010816(n). a(3*n) = A133079(n).

%e G.f. = 1 + 3*x - 5*x^3 - 7*x^6 + 9*x^10 + 11*x^15 - 13*x^21 - 15*x^28 + ...

%e G.f. = q + 3*q^9 - 5*q^25 - 7*q^49 + 9*q^81 + 11*q^121 - 13*q^169 + ...

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3, {x, 0, n}]; (* _Michael Somos_, Jun 19 2015 *)

%o (PARI) {a(n) = if( n<0, 0, if( issquare( 8*n+1, &n), (-1)^( (n-1) \ 4) * n))};

%o (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)))^3, n))};

%Y Cf. A010816, A133079.

%K sign

%O 0,2

%A _Michael Somos_, Sep 09 2007