A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.
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, 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, 6, 0, 0, 0, 5, 6, 4, 2, 2, 0, 0, 0, 4, 7, 6, 2, 0, 8, 2
Offset: 0
Keywords
Examples
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 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- Michael Somos, Introduction to Ramanujan theta functions, 2019.
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Crossrefs
Programs
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Magma
A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 3*A[2] + 3*A[3];
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Mathematica
a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {3, 0, -1, 0, 1, 0, -3, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]]; a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}]; -
PARI
{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 3, 0, -1, 0, 1, 0, -3][d%8 + 1]))}; -
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))}; -
Sage
A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 3*A[1] + 3*A[2];
Formula
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.
Euler transform of period 8 sequence [3, -3, 1, 0, 1, -3, 3, -2, ...].
Moebius transform is period 8 sequence [3, 0, -1, 0, 1, 0, -3, 0, ...].
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
Comments