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A256574 Expansion of chi(x) * psi(-x^3) * psi(x^48) in powers of x where psi(), chi() are Ramanujan theta functions.

%I A256574 #64 Jul 23 2025 15:12:59
%S A256574 1,1,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,
%T A256574 0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,2,0,0,0,0,0,0,0,1,1,0,0,
%U A256574 0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0
%N A256574 Expansion of chi(x) * psi(-x^3) * psi(x^48) in powers of x where psi(), chi() are Ramanujan theta functions.
%C A256574 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A256574 G. C. Greubel, <a href="/A256574/b256574.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..86 from Michael Somos)
%H A256574 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A256574 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A256574 Expansion of q^(-19/3) * eta(q^2)^2 * eta(q^3) * eta(q^12) * eta(q^96)^2 / (eta(q) * eta(q^4) * eta(q^6) * eta(q^48)) in powers of q.
%F A256574 Euler transform of a period 96 sequence.
%F A256574 2 * a(n) = A257403(3*n + 19) unless n == 2 (mod 8).
%F A256574 a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = a(16*n + 9) = a(16*n + 13) = 0.
%F A256574 a(4*n + 1) = A257402(n). a(8*n) = A255317(n). a(16*n + 1) = A255318(n). a(16*n + 5) = A255319(n).
%F A256574 a(n) = (-1)^n * A255320(n). - _Michael Somos_, Apr 24 2015
%F A256574 Expansion of f(x, x^5) * psi(x^48) in powers of x where psi(), f() are Ramanujan theta functions. - _Michael Somos_, Oct 25 2015
%F A256574 G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263767.
%e A256574 G.f. = 1 + x + x^5 + x^8 + x^16 + x^21 + x^33 + x^40 + x^48 + x^49 + ...
%e A256574 G.f. = q^19 + q^22 + q^34 + q^43 + q^67 + q^82 + q^118 + q^139 + q^163 + ...
%t A256574 a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 2, 0, x^24] / (2^(3/2) x^(51/8)), {x, 0, n}];
%t A256574 a[ n_] := If[ n < 0 || Mod[n, 8] == 2, 0, (1/2) Times @@ (Which[# < 5, Boole[# + #2 == 3], Mod[#, 8] > 4, Mod[#2 + 1, 2], True, #2 + 1] & @@@ FactorInteger[ 3 n + 19])]; (* _Michael Somos_, Oct 25 2015 *)
%o A256574 (PARI) {a(n) = my(A, p, e); if( n<0 || n%8 == 2, 0, A = factor(3*n + 19); 1/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p+e==3, p%8 > 4, 1-e%2, e+1)))};
%o A256574 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) * eta(x^96 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^48 + A)), n))};
%Y A256574 Cf. A255317, A255318, A255319, A255320, A257402, A257403, A263767.
%K A256574 nonn
%O A256574 0,57
%A A256574 _Michael Somos_, Apr 22 2015