cp's OEIS Frontend

A143378 Expansion of q^(-1/24) * eta(q^2)^5 / (eta(q) * eta(q^4)^2) in powers of q.

%I A143378 #14 Feb 16 2025 08:33:08
%S A143378 1,1,-3,-2,2,-1,0,1,2,4,-2,0,-1,-2,2,-1,0,-2,-2,-2,0,0,1,4,-2,2,1,0,
%T A143378 -2,0,4,0,2,0,0,1,0,-4,0,-2,-3,0,2,2,-4,0,0,2,-2,0,-2,-3,2,0,2,2,0,1,
%U A143378 4,0,0,0,2,0,0,-4,0,2,0,2,-1,0,0,2,-2,2,-2,-1,-2,-4,0,0,0,-2,-2,0,0,2,2,-2,2,0,1,0,0,-2,0,0,0,-2,5,2,-4,2,0
%N A143378 Expansion of q^(-1/24) * eta(q^2)^5 / (eta(q) * eta(q^4)^2) in powers of q.
%C A143378 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A143378 G. C. Greubel, <a href="/A143378/b143378.txt">Table of n, a(n) for n = 0..1000</a>
%H A143378 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A143378 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A143378 Expansion of phi(x) * f(-x) = phi(-x^2) * f(x) = chi(-x) * f(x)^2 = phi(-x^2)^2 / chi(-x) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%F A143378 Euler transform of period 4 sequence [ 1, -4, 1, -2, ...].
%F A143378 G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 288^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143380.
%F A143378 G.f.: Product_{k>0} (1 - (-x)^k)^2 * (1 - x^(2*k-1)).
%F A143378 a(n) = A143377(4*n) = A143380(4*n).
%e A143378 G.f. = 1 + x - 3*x^2 - 2*x^3 + 2*x^4 - x^5 + x^7 + 2*x^8 + 9*x^9 + ...
%e A143378 G.f. = q + q^25 - 3*q^49 - 2*q^73 + 2*q^97 - q^121 + q^169 + 2*q^193 + ...
%t A143378 a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 3, 0, x], {x, 0, n}]; (* _Michael Somos_, Apr 07 2015 *)
%t A143378 a[ n_] := SeriesCoefficient[ QPochhammer[ -x] EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* _Michael Somos_, Apr 07 2015 *)
%o A143378 (PARI) {a(n) = my(A, p, e, x); if(n<0, 0, n *= 4; A = factor(6*n + 1); simplify( I^n * prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if(p<5, 0, if(p%8==5 || p%24==23, !(e%2), if(p%8==3 || p%24==17, (-1)^(e\2) * !(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e)))))))};
%o A143378 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A)^2), n))};
%Y A143378 Cf. A143377, A143380.
%K A143378 sign
%O A143378 0,3
%A A143378 _Michael Somos_, Aug 11 2008