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 A266575 Expansion of q * f(-q^4)^6 / phi(-q) in powers of q where phi(), f() are Ramanujan theta functions. 2
 1, 2, 4, 8, 8, 12, 16, 16, 25, 28, 28, 32, 40, 40, 48, 64, 48, 62, 76, 64, 80, 92, 80, 96, 121, 100, 112, 128, 120, 136, 160, 128, 144, 184, 152, 200, 200, 164, 208, 224, 192, 216, 252, 224, 248, 296, 224, 256, 337, 262, 312, 320, 280, 336, 368, 320, 336, 396 (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..2500 Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of q * phi(q) * psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. Expansion of eta(q^2) * eta(q^4)^6 / eta(q)^2 in powers of q. Euler transform of period 4 sequence [2, 1, 2, -5, ...]. G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(-3/2) (t/I)^(5/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A245643. G.f.: x * Product_{k>0} (1 + x^k) * (1 - x^(4*k))^6 / (1 - x^k). Convolution inverse of A134414. EXAMPLE G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 16*x^8 + ... MATHEMATICA a[ n_] := SeriesCoefficient[ q QPochhammer[ q^4]^6 / EllipticTheta[ 4, 0, q], {q, 0, n}]; a[ n_] := SeriesCoefficient[ 2^-4 EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q]^4, {q, 0, n}]; PROG (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^6 / eta(x + A)^2, n))}; (MAGMA) A := Basis( ModularForms( Gamma1(4), 5/2), 59); A[2]; CROSSREFS Cf. A134414. A245643. Sequence in context: A078750 A054785 A236924 * A260514 A123263 A008218 Adjacent sequences:  A266572 A266573 A266574 * A266576 A266577 A266578 KEYWORD nonn AUTHOR Michael Somos, Jan 03 2016 STATUS approved

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Last modified August 15 00:32 EDT 2020. Contains 336484 sequences. (Running on oeis4.)