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A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A244540 #21 Jun 08 2025 02:28:13
%S A244540 1,3,3,2,3,4,2,0,3,5,4,2,2,4,0,0,3,6,5,2,4,0,2,0,2,7,4,4,0,4,0,0,3,4,
%T A244540 6,0,5,4,2,0,4,6,0,2,2,4,0,0,2,3,7,4,4,4,4,0,0,4,4,2,0,4,0,0,3,8,4,2,
%U A244540 6,0,0,0,5,6,4,2,2,0,0,0,4,7,6,2,0,8,2
%N A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A244540 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A244540 G. C. Greubel, <a href="/A244540/b244540.txt">Table of n, a(n) for n = 0..2500</a>
%H A244540 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A244540 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A244540 Expansion of f(-q^3, -q^5)^2 * phi(q) / psi(-q) = f(-q^3, -q^5)^2 * chi(q)^3 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A244540 Euler transform of period 8 sequence [3, -3, 1, 0, 1, -3, 3, -2, ...].
%F A244540 Moebius transform is period 8 sequence [3, 0, -1, 0, 1, 0, -3, 0, ...].
%F A244540 Convolution product of A244526 and A107635. Convolution product of A000122 and A093709.
%F A244540 a(n) = (A004018(n) + A033715(n)) / 2 = A244543(2*n).
%F A244540 a(2*n) = a(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 4*A053692(n). a(8*n + 7) = 0.
%F A244540 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 + 1/sqrt(2))/2 = 2.681517... . - _Amiram Eldar_, Jun 08 2025
%e A244540 G.f. = 1 + 3*q + 3*q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 2*q^6 + 3*q^8 + 5*q^9 + ...
%t A244540 a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {3, 0, -1, 0, 1, 0, -3, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
%t A244540 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];
%o A244540 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 3, 0, -1, 0, 1, 0, -3][d%8 + 1]))};
%o A244540 (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 * (A + subst(A, x, x^2)) / 2, n))};
%o A244540 (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 3*A[1] + 3*A[2];
%o A244540 (Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 3*A[2] + 3*A[3];
%Y A244540 Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A093709, A107635, A121373, A244526, A244543.
%K A244540 nonn
%O A244540 0,2
%A A244540 _Michael Somos_, Jun 29 2014