cp's OEIS Frontend

A159818 Expansion of f(q) * f(q^5) in powers of q where f() is a Ramanujan theta function.

%I A159818 #20 Feb 16 2025 08:33:10
%S A159818 1,1,-1,0,0,0,1,-2,0,0,-2,-1,-1,0,0,2,0,0,0,0,1,0,2,0,0,-2,0,2,0,0,1,
%T A159818 1,0,0,0,0,-2,2,0,0,0,0,1,0,0,-2,0,0,0,0,0,-2,0,0,0,0,-1,-2,0,0,-2,-1,
%U A159818 0,0,0,2,0,2,0,0,-2,0,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-2,0,0,1,0,2,0,0,0,0,2,0,0,2,1,-2,0,0
%N A159818 Expansion of f(q) * f(q^5) in powers of q where f() is a Ramanujan theta function.
%C A159818 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A159818 Number 71 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A159818 Seiichi Manyama, <a href="/A159818/b159818.txt">Table of n, a(n) for n = 0..10000</a>
%H A159818 Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A159818 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A159818 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A159818 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A159818 Expansion of q^(-1/4) * eta(q^2)^3 * eta(q^10)^3 / (eta(q) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.
%F A159818 Euler transform of period 20 sequence [ 1, -2, 1, -1, 2, -2, 1, -1, 1, -4, 1, -1, 1, -2, 2, -1, 1, -2, 1, -2, ...].
%F A159818 a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*z) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2 and z = 1 if x or y == 0 (mod 4) else z = 0.
%F A159818 G.f. is a period 1 Fourier series which satisfies f(-1 / (320 t)) = (320)^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
%F A159818 G.f.: Product_{k>0} (1 - (-x)^k) * (1 - (-x)^(5*k)).
%F A159818 a(n) = (-1)^n * A030202(n). Convolution square is A159817. a(5*n + 3) = a(5*n + 4) = a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -a(n).
%e A159818 G.f. = 1 + x - x^2 + x^6 - 2*x^7 - 2*x^10 - x^11 - x^12 + 2*x^15 + x^20 + ...
%e A159818 G.f. = q + q^5 - q^9 + q^25 - 2*q^29 - 2*q^41 - q^45 - q^49 + 2*q^61 + q^81 + ...
%t A159818 a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^5], {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o A159818 (PARI) {a(n) = my(A, p, e, x, z); if(n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if(p==2, 0, p==5, 1, p%20>10, !(e%2), p%4==3, kronecker(-4, e+1), for(y=1, sqrtint(p\5), if(issquare(p - 5*y^2, &x), z = if(x%2, y, x)%4/2; break)); (-1)^(e*z) *(e+1))))};
%o A159818 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^10 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};
%Y A159818 Cf. A030202, A159817.
%K A159818 sign
%O A159818 0,8
%A A159818 _Michael Somos_, Apr 22 2009