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A058700
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Coefficients of replicable function number 49a.
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2
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1, 0, 2, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 19, 25, 28, 35, 41, 50, 58, 71, 81, 98, 113, 134, 154, 183, 208, 244, 280, 326, 371, 431, 489, 565, 641, 735, 832, 953, 1075, 1225, 1382, 1569, 1764, 1999, 2243, 2533, 2839, 3195, 3575, 4018, 4484, 5026, 5604, 6267
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OFFSET
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-1,3
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LINKS
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FORMULA
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a(n) ~ exp(4*Pi*sqrt(n)/7) / (sqrt(14)*n^(3/4)). - Vaclav Kotesovec, Sep 07 2017
Expansion of A - 3*q, where A = (eta(q^7)^4 + 7*eta(q)^2*eta(q^49)^2)/ (eta(q)*eta(q^49)*(eta(q)^2 + 7*eta(q)*eta(q^49) + 7*eta(q^49)^2)), in powers of q. - G. C. Greubel, Jun 17 2018
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 06 2018
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EXAMPLE
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T49a = 1/q + 2*q + q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 5*q^6 + 7*q^7 + 8*q^8 + ...
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MATHEMATICA
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QP = QPochhammer; A1 = QP[q]; A2 = QP[q^7]; A3 = QP[q^49]; s = -3 q + (A2^4 + 7*q^3*A1^2*A3^2)/(A1*A3)/(A1^2 + 7*q^2*A1*A3 + 7*q^4*A3^2) + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 16 2015, adapted from A136560 *)
a[ n_] := With[ {e1 = QPochhammer[ q], e2 = QPochhammer[ q^7], e3 = QPochhammer[ q^49]}, SeriesCoefficient[ -3 + (e2^4 + 7 q^3 e1^2 e3^2) / (q e1 e3 (e1^2 + 7 q^2 e1 e3 + 7 q^4 e3^2)), {q, 0, n}]]; (* Michael Somos, Sep 06 2018 *)
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PROG
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(PARI) q='q+O('q^50); A = (eta(q^7)^4 + 7*q^3*(eta(q)*eta(q^49))^2)/( eta(q)*eta(q^49)*(eta(q)^2 + 7*q^2*eta(q)*eta(q^49) + 7*q^4* eta(q^49)^2)); Vec(A-3*q) \\ G. C. Greubel, Jun 17 2018
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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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