A132004 - OEIS

%I #26 Mar 12 2021 22:24:44

%S 1,-1,1,-1,2,-1,0,-1,1,-2,0,-1,2,0,2,-1,2,-1,0,-2,0,0,0,-1,3,-2,1,0,2,

%T -2,0,-1,0,-2,0,-1,2,0,2,-2,2,0,0,0,2,0,0,-1,1,-3,2,-2,2,-1,0,0,0,-2,

%U 0,-2,2,0,0,-1,4,0,0,-2,0,0,0,-1,2,-2,3,0,0,-2

%N Expansion of (1 - phi(q^3) / phi(q) * phi(-q^2) * phi(-q^6)) / 2 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 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

%H G. C. Greubel, <a href="/A132004/b132004.txt">Table of n, a(n) for n = 1..1000</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 (1 - eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3)) / 2 in powers of q.

%F a(n) is multiplicative with a(2^e) = 2*0^e - 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).

%F G.f.: Sum_{k>0} x^k / (1 + x^k) * Kronecker(-36, k).

%F a(3*n) = a(n). -2 * a(n) = A132003(n) unless n = 0. a(2*n) = - A035154(n). a(2*n + 1) = A125079(n).

%F a(n) = (-1)^n * A035154(n). a(12*n + 7) = a(12*n + 11) = 0. - _Michael Somos_, Nov 01 2015

%F a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 2) = - A125079(n). - _Michael Somos_, Nov 01 2015

%F a(6*n) = - A035154(n). a(6*n + 2) = - A122865(n). a(6*n + 4) = - A122856(n). - _Michael Somos_, Nov 01 2015

%F a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). - _Michael Somos_, Nov 01 2015

%e G.f. = x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...

%t a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* _Michael Somos_, Nov 01 2015 *)

%t a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Nov 01 2015 *)

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(n+d) * kronecker( -36, d)))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3)) / 2, 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==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

%Y Cf. A008441, A035154, A053692, A122856, A122865, A125079, A132003, A258277, A258278.

%K sign,mult

%O 1,5

%A _Michael Somos_, Aug 06 2007