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A092833
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Expansion of q / (chi(-q) * chi(-q^23)) in powers of q where chi() is a Ramanujan theta function.
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3
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1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 105, 123, 143, 167, 194, 225, 260, 301, 346, 398, 458, 524, 600, 686, 782, 891, 1014, 1151, 1306, 1480, 1674, 1892, 2137, 2409, 2713, 3053, 3431, 3852, 4322, 4842, 5421, 6064, 6776
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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G.f.: x * (Product_{k>0} (1 + x^k) * (1 + x^(23*k))).
Expansion of eta(q^2) * eta(q^46) / (eta(q) * eta(q^23)) in powers of q.
Euler transform of period 46 sequence with g.f. x / (1 - x^2) + x^23 / (1 - x^46).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 2 * u*v * (1 + v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (46 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132322.
a(n) ~ exp(2*Pi*sqrt(2*n/23)) / (2^(7/4) * 23^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
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EXAMPLE
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G.f. = q + q^2 + q^3 + 2*q^4 + 2*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + 8*q^10 + ...
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MATHEMATICA
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a[n_] := Coefficient[ Series[ x*Product[(1 + x^k)*(1 + x^(23*k)), {k, 1, n}], {x, 0, n}], x, n]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jan 28 2013, from 1st formula *)
a[ n_] := SeriesCoefficient[ q Product[ (1 + q^k) (1 + q^(23 k)), {k, n}], {q, 0, n}]; (* Michael Somos, Aug 11 2015 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ -q, q] QPochhammer[ -q^23, q^23]), {q, 0, n}]; (* Michael Somos, Aug 11 2015 *)
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PROG
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(PARI) {a(n) = my(A, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = A + A^2 + sqrt(A + (A + A^2)^2)); polcoeff(A, n))};
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^46 + A) / eta(x + A) / eta(x^23 + A), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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