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A331779
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Coefficients arising from a 4-dissection of the Rogers-Ramanujan functions.
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1
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2, 1, 3, 2, 3, 0, 2, 0, 2, 0, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 0, 2, 1, 2, 0, 3, 1, 3, 1, 2, 0, 2, 1, 3, 2, 3, 0, 2, 0, 2, 0, 3, 2, 3, 1, 2, 0, 2, 1, 3, 1, 3, 0, 2, 1, 2, 0, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 0, 2, 0, 2, 0, 3, 2, 3, 1, 2, 0
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OFFSET
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1,1
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COMMENTS
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Periodic with period 80.
a(n) = 0 if n = 0, +-6, +-8, +-10, +-22, +-26, +-32, +-38, 40 (mod 80).
a(n) = 1 if n = +-2, +-12, +-14, +-16, +-18, +-20, +-24, +-28, +-30, +-34 (mod 80).
a(n) = 2 if n = +-1, +-4, +-7, +-9, +-15, +-17, +-23, +-25, +-31, +-33, +-36, +-39 (mod 80).
a(n) = 3 if n = +-3, +-5, +-11, +-13, +-19, +-21, +-27, +-29, +-35, +-37 (mod 80). (End)
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,1).
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FORMULA
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G.f.: x*(1 - x + x^2)*(2 + 3*x + 4*x^2 + 3*x^3 + 4*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 7*x^8 + 5*x^9 + 6*x^10 + 4*x^11 + 8*x^12 + 6*x^13 + 8*x^14 + 6*x^15 + 8*x^16 + 4*x^17 + 6*x^18 + 5*x^19 + 7*x^20 + 3*x^21 + 3*x^22 + 2*x^23 + 4*x^24 + 3*x^25 + 4*x^26 + 3*x^27 + 2*x^28) / ((1 - x)*(1 + x)*(1 + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4)*(1 - x^2 + x^4 - x^6 + x^8)*(1 + x^8)).
a(n) = -a(n-4) - a(n-8) - a(n-12) + a(n-20) + a(n-24) + a(n-28) + a(n-32) for n>32.
(End)
Product_{k>=1} 1/(1-q^k)^a(k) = ( (q^6, q^8, q^10, q^16; q^16)_{oo} * (q^2, q^6, q^8, q^10, q^12, q^14, q^18, q^20; q^20)_{oo} )/( (q; q)_{oo}^2 * (q^3, q^5; q^8)_{oo} ), where (a; q)_{oo} is the q-Pochhammer symbol and (a, b; q)_{oo} = (q;a)_{oo}*(b;q)_{oo} (See Eqs (5) and (13) of Hirschhorn ref.). - G. C. Greubel, Feb 10 2020
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MATHEMATICA
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g[x_]:= (2 +3*x +4*x^2 +3*x^3 +4*x^4 +2*x^5 +3*x^6 +3*x^7 +7*x^8 +5*x^9 +6*x^10 +4*x^11 +8*x^12 +6*x^13 +8*x^14 +6*x^15 +8*x^16 +4*x^17 +6*x^18 +5*x^19 +7*x^20 +3*x^21 +3*x^22 +2*x^23 +4*x^24 +3*x^25 +4*x^26 +3*x^27 +2*x^28); CoefficientList[Series[x*(1+x^3)*(1-x^4)*g[x]/((1+x)*(1-x^16)*(1-x^20)), {x, 0, 100}], x] (* G. C. Greubel, Feb 10 2020 *)
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PROG
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(PARI) Vec(x*(1 - x + x^2)*(2 + 3*x + 4*x^2 + 3*x^3 + 4*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 7*x^8 + 5*x^9 + 6*x^10 + 4*x^11 + 8*x^12 + 6*x^13 + 8*x^14 + 6*x^15 + 8*x^16 + 4*x^17 + 6*x^18 + 5*x^19 + 7*x^20 + 3*x^21 + 3*x^22 + 2*x^23 + 4*x^24 + 3*x^25 + 4*x^26 + 3*x^27 + 2*x^28) / ((1 - x)*(1 + x)*(1 + x^2)*(1 - x + x^2 - x^3 + x^4)*(1 + x^4)*(1 + x + x^2 + x^3 + x^4)*(1 - x^2 + x^4 - x^6 + x^8)*(1 + x^8)) + O(x^40)) \\ Colin Barker, Feb 10 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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