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A123532
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Expansion of (eta(q) * eta(q^6))^7 / (eta(q^2) * eta(q^3))^5 in powers of q.
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3
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1, -7, 19, -23, 6, 11, 8, -55, 73, -42, 12, -5, 14, -56, 114, -119, 18, 65, 20, -138, 152, -84, 24, -37, 31, -98, 235, -184, 30, 66, 32, -247, 228, -126, 48, 49, 38, -140, 266, -330, 42, 88, 44, -276, 438, -168, 48, -101, 57, -217, 342, -322, 54, 227, 72, -440, 380, -210, 60, -30, 62, -224
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Euler transform of period 6 sequence [ -7, -2, -2, -2, -7, -4, ...].
Expansion of q * (phi(-q) * psi(q^3))^3 / (phi(-q^3) * psi(q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (b(q)^2 / b(q^2)) * (c(q^2)^2 / c(q)) / 3 in powers of q where b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 6 (t / i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 19 2013
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 -x^(3*k))^2 * (1 + x^(3*k))^7 / (1 + x^k)^5.
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EXAMPLE
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G.f. = q - 7*q^2 + 19*q^3 -23*q^4 + 6*q^5 + 11*q^6 + 8*q^7 - 55*q^8 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Sum[ { 1, -4, 6, -4, 1, 0}[[ Mod[ d, 6, 1]]] d, {d, Divisors[ n]}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^6])^7 / (QPochhammer[ q^2] QPochhammer[ q^3])^5, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d*[ 0, 1, -4, 6, -4, 1][ d%6 + 1]))};
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A))^7 / (eta(x^2 + A) * eta(x^3 + A))^5, n))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, n/d *[ 0, 1, -9, 16, -9, 1][ d%6 + 1]))}; /* Michael Somos, Sep 19 2013 */
(Sage) A = ModularForms( Gamma0(6), 2, prec=50) . basis(); A[1] - 7*A[2]; # Michael Somos, Sep 19 2013
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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