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 A204386 Expansion of (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2. 1
 1, 12, 28, 96, 126, 336, 344, 768, 757, 1512, 1332, 2688, 2198, 4128, 3528, 6144, 4914, 9084, 6860, 12096, 9632, 15984, 12168, 21504, 15751, 26376, 20440, 33024, 24390, 42336, 29792, 49152, 37296, 58968, 43344, 72672, 50654, 82320, 61544, 96768 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 = 1..1000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of x * psi(x)^8 + 4 * x^2 * psi(x^2)^8 in powers of x where psi() is a Ramanujan theta function. Expansion of (eta(q^2)^2 / eta(q))^8 + 4 * (eta(q^4)^2 / eta(q^2))^8 in powers of q. a(n) is multiplicative with a(2^e) = 3/2 * 8^e if e>0, a(p^e) = ((p^3) ^ (e+1) - 1) / (p^3 - 1). a(2*n + 1) = A045823(n). a(2*n) = 12 * A007331(n). Convolution of this sequence with A004018 is A050468. EXAMPLE x + 12*x^2 + 28*x^3 + 96*x^4 + 126*x^5 + 336*x^6 + 344*x^7 + 768*x^8 + ... MATHEMATICA a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^(1/2)]^8 + 4*EllipticTheta[2, 0, q]^8)/256, {q, 0, n}];  Table[a[n], {n, 1, 50}] (* G. C. Greubel, Apr 13 2018 *) CoefficientList[Series[(EllipticTheta[2, 0, q^(1/2)]^8 +4*EllipticTheta[2, 0, q]^8)/ 256, {q, 0, 50}], q] (* Vaclav Kotesovec, Apr 13 2018 *) PROG (PARI) {a(n) = if( n<1, 0, if( n%2, sigma( n, 3), 12 * sumdiv( n/2, d, (n/2/d%2) * d^3)))} (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8 + 4 * x * (eta(x^4 + A)^2 / eta(x^2 + A))^8, n))} CROSSREFS Cf. A004018, A007331, A045823, A050468. Sequence in context: A039366 A043189 A043969 * A078441 A340685 A203026 Adjacent sequences:  A204383 A204384 A204385 * A204387 A204388 A204389 KEYWORD nonn,mult AUTHOR Michael Somos, Jan 15 2012 STATUS approved

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Last modified September 21 21:32 EDT 2021. Contains 347605 sequences. (Running on oeis4.)