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A134461 Expansion of (phi(x) * psi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A134461 #30 Feb 16 2025 08:33:07
%S A134461 1,4,-2,-24,-11,44,22,-8,50,-44,-96,56,-121,-152,198,160,176,48,-162,
%T A134461 88,-198,-52,22,-528,233,200,-242,-88,-176,668,550,264,-44,-188,224,
%U A134461 -728,154,-484,-1056,656,-311,-236,-100,792,714,-528,640,88,-478,-484,1566,968,192,780,-1994,-648,-942
%N A134461 Expansion of (phi(x) * psi(-x))^4 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A134461 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A134461 Number 34 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A134461 Seiichi Manyama, <a href="/A134461/b134461.txt">Table of n, a(n) for n = 0..10000</a>
%H A134461 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 A134461 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H A134461 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A134461 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A134461 Expansion of q^(-1/2) * (eta(q^2)^4 / (eta(q) * eta(q^4)))^4 in powers of q.
%F A134461 Euler transform of period 4 sequence [ 4, -12, 4, -8, ...].
%F A134461 a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = b(p)*b(p^(e-1)) - p^3*b(p^(e-2)).
%F A134461 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 256 (t/i)^4 f(t) where q = exp(2 Pi i t).
%F A134461 G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2 / (1 + x^(2*k)))^4.
%F A134461 a(n) = (-1)^n * A030211(n).
%F A134461 Convolution square is A216711. - _Michael Somos_, Jun 10 2015
%e A134461 G.f. = 1 + 4*x - 2*x^2 - 24*x^3 - 11*x^4 + 44*x^5 + 22*x^6 - 8*x^7 + ...
%e A134461 G.f. = q + 4*q^3 - 2*q^5 - 24*q^7 - 11*q^9 + 44*q^11 + 22*q^13 - 8*q^15 + ...
%t A134461 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^2] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)))^4, {x, 0, n}]; (* _Michael Somos_, Jun 10 2015 *)
%o A134461 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)) )^4, n))};
%o A134461 (Magma) A := Basis( CuspForms( Gamma0(16), 4), 115); A[1] + 4*A[3]; /* _Michael Somos_, Jun 10 2015 */
%Y A134461 Cf. A216711.
%Y A134461 The same as A030211 except for signs.
%K A134461 sign
%O A134461 0,2
%A A134461 _Michael Somos_, Oct 26 2007