A129134 - OEIS

%I #22 Sep 20 2024 21:35:50

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

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

%U 0,0,-1,0,4,-2,-2,0,0,0,-3,2,0,-2,-2,0,0,0,0,5,2,-2,0,0,2,0,-2,2,0,0,0,0,0,0,-2,2,1,-6,-1,0,4,0,0,0

%N Expansion of (1 - phi(-q) * phi(-q^2)) / 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).

%C For n nonzero, a(n) is nonzero if and only if n is in A002479.

%H G. C. Greubel, <a href="/A129134/b129134.txt">Table of n, a(n) for n = 1..5000</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^2) / eta(q^4)) / 2 in powers of q.

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

%F a(n) = A002325(n) * (-1)^floor((n-1)/2). A082564(n) = -2 * a(n) unless n=0.

%F a(3*n + 1) = A258747(n). a(3*n + 2) = A258764(n). - _Michael Somos_, Jun 09 2015

%e G.f. = q + q^2 - 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 2*q^11 - 2*q^12 - q^16 + ...

%t a[ n_] := If[ n < 1, 0, (-1)^Quotient[ n - 1, 2] DivisorSum[n, KroneckerSymbol[-2, #] &]]; (* _Michael Somos_, Jun 09 2015 *)

%t a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2]) / 2 , {q, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)

%t a[ n_] := SeriesCoefficient[ (1 - QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4]) / 2 , {q, 0, n}]; (* _Michael Somos_, Jun 09 2015 *)

%o (PARI) {a(n) = if( n<1, 0, (-1)^((n-1)\2) * sumdiv(n, d, kronecker( -2, d)))};

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

%Y Cf. A002325, A002479, A082564, A258747, A258764.

%K sign

%O 1,3

%A _Michael Somos_, Mar 30 2007