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A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.

%I A244544 #18 Feb 16 2025 08:33:23
%S A244544 1,2,3,2,3,2,2,0,3,4,4,2,2,2,0,0,3,4,5,2,4,0,2,0,2,4,4,4,0,2,0,0,3,4,
%T A244544 6,0,5,2,2,0,4,4,0,2,2,2,0,0,2,2,7,4,4,2,4,0,0,4,4,2,0,2,0,0,3,4,4,2,
%U A244544 6,0,0,0,5,4,4,2,2,0,0,0,4,6,6,2,0,4,2
%N A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.
%C A244544 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A244544 G. C. Greubel, <a href="/A244544/b244544.txt">Table of n, a(n) for n = 0..10000</a>
%H A244544 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A244544 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A244544 Expansion of f(-q^3, -q^5)^4 / psi(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.
%F A244544 Euler transform of period 8 sequence [ 2, 0, -2, 2, -2, 0, 2, -2, ...].
%F A244544 Moebius transform is period 8 sequence [ 2, 1, 0, 0, 0, -1, -2, 0, ...].
%F A244544 Convolution square of A093709.
%F A244544 a(2*n) = A244540(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 2*A053692(n). a(8*n + 7) = 0.
%e A244544 G.f. = 1 + 2*q + 3*q^2 + 2*q^3 + 3*q^4 + 2*q^5 + 2*q^6 + 3*q^8 + 4*q^9 + ...
%t A244544 a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {2, 1, 0, 0, 0, -1, -2, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
%t A244544 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];
%o A244544 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 2, 1, 0, 0, 0, -1, -2][d%8 + 1]))};
%o A244544 (PARI) {a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( (A + subst(A, x, x^2))^2 / 4, n))};
%o A244544 (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 2*A[1] + 3*A[2];
%o A244544 (Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 2*A[2] + 3*A[3];
%Y A244544 Cf. A033761, A053692, A093709, A244540.
%K A244544 nonn
%O A244544 0,2
%A A244544 _Michael Somos_, Jun 29 2014