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 A216711 Expansion of q * (phi(q) * psi(-q))^8 in powers of q where phi(), psi() are Ramanujan theta functions. 2
 1, 8, 12, -64, -210, 96, 1016, 512, -2043, -1680, 1092, -768, 1382, 8128, -2520, -4096, 14706, -16344, -39940, 13440, 12192, 8736, 68712, 6144, -34025, 11056, -50760, -65024, -102570, -20160, 227552, 32768, 13104, 117648, -213360, 130752, 160526, -319520 (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 Seiichi Manyama, Table of n, a(n) for n = 1..10000 Michael Somos, Introduction to Ramanujan theta functions Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q. a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2. G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t). a(n) = -(-1)^n * A002288(n). Convolution square of A134461. EXAMPLE G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}]; PROG (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))}; (MAGMA) A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* Michael Somos, Jun 10 2015 */ CROSSREFS Cf. A002288, A134461. Sequence in context: A166625 A038290 A002288 * A137232 A147764 A226259 Adjacent sequences:  A216708 A216709 A216710 * A216712 A216713 A216714 KEYWORD sign,mult,look AUTHOR Michael Somos, Apr 10 2013 STATUS approved

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Last modified July 6 04:31 EDT 2022. Contains 355108 sequences. (Running on oeis4.)