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#27 by Bruno Berselli at Thu Oct 18 03:06:03 EDT 2018
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#26 by Michel Marcus at Thu Oct 18 01:09:08 EDT 2018
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#25 by Michel Marcus at Thu Oct 18 01:08:57 EDT 2018
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| LINKS
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Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
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| STATUS
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approved
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#24 by N. J. A. Sloane at Thu Nov 02 15:38:29 EDT 2017
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#23 by Robert Israel at Thu Nov 02 15:37:40 EDT 2017
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#22 by Robert Israel at Thu Nov 02 15:37:32 EDT 2017
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| MAPLE
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N:= 100: # to get a(1)..a(N)
S:= series(q*Product(1-q^(9*k), k=1..N/9)/Product((1-q^k)^3, k=1..N), q, N+1):
seq(coeff(S, q, n), n=1..N); # Robert Israel, Nov 02 2017
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| STATUS
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approved
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#21 by N. J. A. Sloane at Thu Nov 02 15:34:27 EDT 2017
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#20 by Peter Luschny at Thu Nov 02 14:57:11 EDT 2017
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#19 by Michael Somos at Thu Nov 02 14:48:25 EDT 2017
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#18 by Michael Somos at Thu Nov 02 14:47:55 EDT 2017
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| NAME
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ExpansionSeries expansion of (eta(q^9)/) / eta(q))^3 in powers of q.
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| COMMENTS
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Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
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| FORMULA
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Euler transform of period 9 sequence [ [3, 3, 3, 3, 3, 3, 3, 3, 0, ...].
G.f.: x*( * (Product_{k>0} (1- - x^(9*k))/() / (1- - x^k))^3.
Expansion of c(q^3) / (3 * b(q)) = (c(q) / (3 * b(q)) ^3))^3 in powers of q where b(), c() are cubic AGM functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/27) / f) g(t) where q = exp(2 Pi i t).) and g() is the g.f. of A131986.
Convolution inverse of A131986. Convolution cube of A104502. - Michael Somos, Nov 02 2017
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| EXAMPLE
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G.f. = q + 3*q^2 + 9*q^3 + 22*q^4 + 51*q^5 + 108*q^6 + 221*q^7 + 429*q^8 + ...
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| MATHEMATICA
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a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^9] / QPochhammer[ q])^3, {q, 0, n}]; (* Michael Somos, Nov 02 2017 *)
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| PROG
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(PARI) {a(n)=local) = my(A); if(( n<1, 0, n--; A= = x* * O(x^n); polcoeff( (eta(x^9+ + A)/) / eta(x+ + A))^3, n))}))};
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| CROSSREFS
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Cf. A104502, A131986.
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| STATUS
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approved
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Discussion
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| Thu Nov 02
| 14:48
| Michael Somos: Added more info. Light and space edits.
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