A331779 Coefficients arising from a 4-dissection of the Rogers-Ramanujan functions.

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

Offset

1,1

Comments

Periodic with period 80.

From G. C. Greubel, Feb 09 2020: (Start)

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)

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Michael D. Hirschhorn, On the 2-and 4-dissections of the Rogers-Ramanujan functions, The Ramanujan Journal 40.2 (2016): 227-235. See Eqs. 14 and 15.

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).

Formula

From Colin Barker, Feb 09 2020: (Start)

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

Mathematica

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 *)

Prog

(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

Crossrefs

Sequence in context: A144154 A350768 A054710 * A243556 A048233 A287730

Adjacent sequences: A331776 A331777 A331778 * A331780 A331781 A331782

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 09 2020

Status

approved