A131947 - OEIS

%I #24 Sep 12 2023 02:28:37

%S 1,-1,1,-5,6,-1,8,-13,1,-6,12,-5,14,-8,6,-29,18,-1,20,-30,8,-12,24,

%T -13,31,-14,1,-40,30,-6,32,-61,12,-18,48,-5,38,-20,14,-78,42,-8,44,

%U -60,6,-24,48,-29,57,-31,18,-70,54,-1,72,-104,20,-30,60,-30,62

%N Expansion of (1 - (phi(-q) * phi(-q^3))^2)/4 in powers of q where phi() is a Ramanujan theta function.

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

%D Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.66).

%H G. C. Greubel, <a href="/A131947/b131947.txt">Table of n, a(n) for n = 1..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 a(n) is multiplicative with a(2^e) = 3 - 2^(e+1), a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.

%F G.f.: Sum_{k>0} k * (-x)^k / (1 - x^k) * Kronecker(9, k) = ((theta_3(-x) * theta_3(-x^3))^2 - 1) / 4.

%F a(n) = -(-1)^n * A113262(n). -4 * a(n) = A131946(n) unless n=0.

%F Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s). - _Amiram Eldar_, Sep 12 2023

%e G.f. = x - x^2 + x^3 - 5*x^4 + 6*x^5 - x^6 + 8*x^7 - 13*x^8 + x^9 - 6*x^10 + ...

%t a[ n_] := SeriesCoefficient[ (1 - (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3])^2) / 4, {q, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)

%t a[ n_] := SeriesCoefficient[ (1 - (QPochhammer[ q] QPochhammer[ q^3])^4 / (QPochhammer[ q^2] QPochhammer[ q^6])^2) / 4, {q, 0, n}]; (* _Michael Somos_, Nov 11 2015 *)

%t a[ n_] := If[ n < 1, 0, Sum[ d {0, 1, -1, 0, -1, 1}[[Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* _Michael Somos_, Nov 11 2015 *)

%t a[ n_] := If[ n < 1, 0, Sum[ n/d {6, 1, -3, -2, -3, 1}[[Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* _Michael Somos_, Nov 11 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*((abs(d%6-3) == 2) - (abs(d%6-3) == 1))))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - (eta(x + A) * eta(x^3 + A))^4 / (eta(x^2 + A) * eta(x^6 + A))^2) / 4, n))};

%o (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3 - p^(e+1), p==3, 1, (p^(e+1) - 1) / (p-1) )))};

%Y Cf. A113262, A131946.

%Y Cf. A000122, A000700, A010054, A121373.

%K sign,easy,mult

%O 1,4

%A _Michael Somos_, Jul 30 2007