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A244543 Expansion of phi(q^2) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

%I A244543 #17 Jun 08 2025 02:29:48
%S A244543 1,1,3,2,3,0,2,0,3,3,4,2,2,0,0,0,3,2,5,2,4,0,2,0,2,1,4,4,0,0,0,0,3,4,
%T A244543 6,0,5,0,2,0,4,2,0,2,2,0,0,0,2,1,7,4,4,0,4,0,0,4,4,2,0,0,0,0,3,0,4,2,
%U A244543 6,0,0,0,5,2,4,2,2,0,0,0,4,5,6,2,0,0,2
%N A244543 Expansion of phi(q^2) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
%C A244543 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A244543 G. C. Greubel, <a href="/A244543/b244543.txt">Table of n, a(n) for n = 0..2500</a>
%H A244543 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>, 2019.
%H A244543 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A244543 Expansion of f(-q^3, -q^5)^2 * phi(q^2) / psi(-q) = f(-q^3, -q^5)^2 * chi(q^2)^2 / chi(-q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
%F A244543 Euler transform of period 8 sequence [1, 2, -1, -2, -1, 2, 1, -2, ...].
%F A244543 Moebius transform is period 8 sequence [1, 2, 1, 0, -1, -2, -1, 0, ...].
%F A244543 a(2*n) = A244540(n). a(2*n + 1) = A113411(n). a(8*n + 1) = A112603(n). a(8*n + 3) = 2* A033761(n). a(8*n + 5) = a(8*n + 7) = 0.
%F A244543 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*(sqrt(2)+1)/4 = 1.896118... . - _Amiram Eldar_, Jun 08 2025
%e A244543 G.f. = 1 + q + 3*q^2 + 2*q^3 + 3*q^4 + 2*q^6 + 3*q^8 + 3*q^9 + 4*q^10 + ...
%t A244543 a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {1, 2, 1, 0, -1, -2, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
%t A244543 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];
%o A244543 (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, 1, 0, -1, -2, -1][d%8 + 1]))};
%o A244543 (PARI) {a(n) = my(A, B); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); B = subst(A, x, x^2); polcoeff( B * (A + B) / 2, n))};
%o A244543 (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + A[1] + 3*A[2];
%o A244543 (Magma) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + A[2] + 3*A[3];
%Y A244543 Cf. A000122, A000700, A010054, A033761, A112603, A113411, A121373, A244540.
%K A244543 nonn
%O A244543 0,3
%A A244543 _Michael Somos_, Jun 29 2014