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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. 2
 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, 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, 6, 0, 0, 0, 5, 2, 4, 2, 2, 0, 0, 0, 4, 5, 6, 2, 0, 0, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). LINKS G. C. Greubel, Table of n, a(n) for n = 0..2500 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA 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. Euler transform of period 8 sequence [1, 2, -1, -2, -1, 2, 1, -2, ...]. Moebius transform is period 8 sequence [1, 2, 1, 0, -1, -2, -1, 0, ...]. 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. EXAMPLE 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 + ... MATHEMATICA a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {1, 2, 1, 0, -1, -2, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]]; a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}]; PROG (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, 1, 0, -1, -2, -1][d%8 + 1]))}; (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))}; (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + A[1] + 3*A[2]; (MAGMA) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + A[2] + 3*A[3]; CROSSREFS Cf. A033761, A112603, A113411, A244540. Sequence in context: A332497 A093347 A230409 * A283979 A134676 A286246 Adjacent sequences:  A244540 A244541 A244542 * A244544 A244545 A244546 KEYWORD nonn AUTHOR Michael Somos, Jun 29 2014 STATUS approved

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Last modified July 29 09:31 EDT 2021. Contains 346344 sequences. (Running on oeis4.)