%I #16 Sep 08 2022 08:45:23
%S 1,-2,-4,8,7,-10,-12,8,18,-18,-16,24,21,-20,-28,32,20,-32,-36,24,42,
%T -42,-28,48,57,-36,-52,40,36,-58,-60,56,48,-66,-48,72,74,-42,-80,80,
%U 61,-82,-72,56,90,-96,-64,72,98,-70,-100,104,64,-106,-108,72,114,-96
%N Expansion of phi(x) * phi(-x)^2 * psi(x^4) 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).
%H G. C. Greubel, <a href="/A113417/b113417.txt">Table of n, a(n) for n = 0..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 q^(-1/2) * (eta(q) * eta(q^8))^2 * (eta(q^2) / eta(q^4))^3 in powers of q.
%F Euler transform of period 8 sequence [ -2, -5, -2, -2, -2, -5, -2, -4, ...].
%F a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
%F G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (2*k - 1) * (-1)^[k/2] * x^(2*k - 1) / (1 - x^(4*k - 2)) = x * (Product_{k>0} ((1 - x^(2*k)) * (1 - x^(4*k)) * (1 + x^(8*k)))^2 / (1 + x^(4*k))).
%F a(n) = (-1)^n * A113419(n) = (-1)^floor(n/2) * A209940(n) = (-1)^(n + floor(n/2)) * A258096(n). - _Michael Somos_, May 19 2015
%F a(n) = A117000(2*n + 1).
%F a(n) = Sum_{d | 2*n + 1} Kronecker(2, d) * d.
%e G.f. = 1 - 2*x - 4*x^2 + 8*x^3 + 7*x^4 - 10*x^5 - 12*x^6 + 8*x^7 + 18*x^8 + ...
%e G.f. = q - 2*q^3 - 4*q^5 + 8*q^7 + 7*q^9 - 10*q^11 - 12*q^13 + 8*q^15 + ...
%t a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* _Michael Somos_, May 19 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^8]^2 QPochhammer[ x^2]^3 / QPochhammer[ x^4]^3, {x, 0, n}]; (* _Michael Somos_, May 19 2015 *)
%o (PARI) {a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, d * (d%2) * (-1)^((d + 1) \ 4)))};
%o (PARI) {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p *= kronecker( 2, p); (p^(e+1) - 1) / (p - 1))))};
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^8 + A))^2 * (eta(x^2 + A) / eta(x^4 + A))^3, n))};
%o (Magma) A := Basis( ModularForms( Gamma1(16), 2), 117); A[2] - 2*A[4] - 4*A[6] + 8*A[8] + 7*A[10] - 10*A[12] - 12*A[14]; /* _Michael Somos_, May 19 2015 */
%Y Cf. A113419, A117000, A209940, A258096.
%K sign
%O 0,2
%A _Michael Somos_, Oct 29 2005