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A178902 Expansion of q^(-1/24) * eta(q^2)^13 / (eta(q)^5 * eta(q^4)^5) in powers of q.

%I A178902 #39 Jun 25 2025 01:03:39
%S A178902 1,5,7,0,0,11,0,-13,0,0,0,0,-17,0,0,-19,0,0,0,0,0,0,-23,0,0,0,25,0,0,
%T A178902 0,0,0,0,0,0,29,0,0,0,0,31,0,0,0,0,0,0,0,0,0,0,35,0,0,0,0,0,-37,0,0,0,
%U A178902 0,0,0,0,0,0,0,0,0,-41,0,0,0,0,0,0,-43,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-47,0,0,0,0,0,0
%N A178902 Expansion of q^(-1/24) * eta(q^2)^13 / (eta(q)^5 * eta(q^4)^5) in powers of q.
%C A178902 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A178902 Seiichi Manyama, <a href="/A178902/b178902.txt">Table of n, a(n) for n = 0..10000</a>
%H A178902 G. Köhler, <a href="http://dx.doi.org/10.7146/math.scand.a-12299">Some eta-identities arising from theta series</a>, Math. Scand. 66 (1990), 147-154.
%H A178902 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A178902 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A178902 Expansion of f(q) * phi(q)^2 = f(q)^3 * chi(q)^2 = phi(q)^3 / chi(q) in powers of q where f(), phi(), chi() are Ramanujan theta functions.
%F A178902 Euler transform of period 4 sequence [5, -8, 5, -3, ...].
%F A178902 a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 13, 17, 19, 23 (mod 24).
%F A178902 G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t).
%F A178902 G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 + x^(2*k - 1))^5 = Sum_{k>0} Kronecker( -6, k) * k * x^((k^2 - 1) / 24) = Sum_{k in Z} (6*k + 1) * (-1)^floor(k/2) * x^(k * (3*k + 1) / 2).
%F A178902 a(n) = (-1)^n * A080332(n).
%e A178902 G.f. = 1 + 5*x + 7*x^2 + 11*x^5 - 13*x^7 - 17*x^12 - 19*x^15 - 23*x^22 + ...
%e A178902 G.f. = q + 5*q^25 + 7*q^49 + 11*q^121 - 13*q^169 - 17*q^289 - 19*q^361 + ...
%t A178902 A178902[n_] := SeriesCoefficient[(QPochhammer[-q, -q]/QPochhammer[q, -q])^3/QPochhammer[-q, q^2], {q, 0, n}]; Table[A178902[n], {n, 0, 50}] (* _G. C. Greubel_, Aug 17 2017 *)
%t A178902 a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m KroneckerSymbol[ -6, m], 0]]; (* _Michael Somos_, Apr 27 2018 *)
%t A178902 a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^13 / (QPochhammer[ x] QPochhammer[ x^4])^5, {x, 0, n}]; (* _Michael Somos_, Apr 27 2018 *)
%o A178902 (PARI) {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -6, n), 0)};
%o A178902 (PARI) {a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, if( p%24<12, p, -p)^(e\2))))};
%o A178902 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 / (eta(x + A)^5 * eta(x^4 + A)^5), n))};
%Y A178902 Apart from signs, same as A080332, A116916, A133079 and A134756.
%K A178902 sign
%O A178902 0,2
%A A178902 _Michael Somos_, Jun 21 2010