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A058490
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Coefficients of replicable function number 12b.
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4
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1, 5, 27, 41, 146, 243, 510, 887, 1755, 2728, 5052, 7857, 13157, 20253, 32805, 48680, 76568, 112320, 169814, 246263, 365013, 519046, 755632, 1063368, 1516404, 2112551, 2972160, 4089098, 5683166, 7750782, 10633276, 14382932, 19539387, 26192432, 35263852
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OFFSET
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0,2
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COMMENTS
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The convolution square of this sequence is A007254 except for the constant term: T12b(q)^2 = T6A(q^2) + 10. - G. A. Edgar, Apr 15 2017
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LINKS
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D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
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FORMULA
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Expansion of q^(1/2) * (eta(q)^3*eta(q^3)^3 / (eta(q^2)^3*eta(q^6)^3) + 8 *eta(q^2)^3*eta(q^6)^3 / (eta(q)^3*eta(q^3)^3)) in powers of q. - G. A. Edgar, Apr 15 2017
Expansion of (chi(-x) * chi(-x^3))^3 + 8*x/(chi(-x) * chi(-x^3))^3 = (chi(-x^3) / chi(-x))^6 - x*(chi(-x) / chi(-x^3))^6 in powers of x.
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = f(t) where q = exp(2 Pi i t).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 13 2017
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EXAMPLE
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T12b = 1/q + 5*q + 27*q^3 + 41*q^5 + 146*q^7 + 243*q^9 + 510*q^11 + ...
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MATHEMATICA
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a[ n_] := With[{A = (QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]))^6}, SeriesCoefficient[ A - x / A, {x, 0, n}]]; (* Michael Somos, Jun 12 2017 *)
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PROG
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(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^6; polcoeff( A - x/A, n))}; /* Michael Somos, Jun 12 2017 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x + A) * eta(x^3 + A) / (eta(x^2 + A) * eta(x^6 + A)))^3; polcoeff( A + 8*x/A, n))}; /* Michael Somos, Jun 12 2017 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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