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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. 1
 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, 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, 6, 0, 0, 0, 5, 4, 4, 2, 2, 0, 0, 0, 4, 6, 6, 2, 0, 4, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of f(-q^3, -q^5)^4 / psi(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions. Euler transform of period 8 sequence [ 2, 0, -2, 2, -2, 0, 2, -2, ...]. Moebius transform is period 8 sequence [ 2, 1, 0, 0, 0, -1, -2, 0, ...]. Convolution square of A093709. a(2*n) = A244540(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 2*A053692(n). a(8*n + 7) = 0. EXAMPLE 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 + ... MATHEMATICA a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {2, 1, 0, 0, 0, -1, -2, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]]; a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}]; PROG (PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, [0, 2, 1, 0, 0, 0, -1, -2][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 + subst(A, x, x^2))^2 / 4, n))}; (Sage) A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 2*A[1] + 3*A[2]; (MAGMA) A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 2*A[2] + 3*A[3]; CROSSREFS Cf. A033761, A053692, A093709, A244540. Sequence in context: A341018 A214323 A321865 * A159580 A121549 A244228 Adjacent sequences:  A244541 A244542 A244543 * A244545 A244546 A244547 KEYWORD nonn AUTHOR Michael Somos, Jun 29 2014 STATUS approved

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Last modified August 4 09:50 EDT 2021. Contains 346446 sequences. (Running on oeis4.)