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A122580
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Number of partitions of n with crank congruent to 0 mod 3, minus number of partitions of n with crank congruent to 1 mod 3.
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1
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1, -2, -1, 3, -1, 1, 2, -3, -2, 3, -3, -1, 5, -4, 0, 5, -3, 0, 7, -8, -3, 9, -6, -2, 9, -10, -3, 13, -11, -1, 15, -13, -3, 18, -14, -3, 22, -20, -7, 27, -21, -3, 29, -27, -8, 34, -30, -7, 42, -37, -8, 48, -39, -9, 55, -50, -13, 66, -52, -11, 74
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OFFSET
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0,2
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COMMENTS
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For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).
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LINKS
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FORMULA
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Expansion of q^(1/24) * eta(q)^2 / eta(q^3) in powers of q. - Michael Somos, Jul 04 2012
G.f.: Product((1-x^n)/(1+x^n+x^(2*n)),n=1..infinity). Euler transform of period 3 sequence [ -2,-2,-1, ...].
a(n) ~ (exp(-Pi*i/9)*exp(-2*Pi*i*n/3) + exp(Pi*i/9)*exp(2*Pi*i*n/3)) * exp(Pi*sqrt(2*n/3)/3) / sqrt(6*n), where i is the imaginary unit [Kane, 2004]. - Vaclav Kotesovec, May 08 2020
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EXAMPLE
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1 - 2*x - x^2 + 3*x^3 - x^4 + x^5 + 2*x^6 - 3*x^7 - 2*x^8 + 3*x^9 - 3*x^10 + ...
1/q - 2*q^23 - q^47 + 3*q^71 - q^95 + q^119 + 2*q^143 - 3*q^167 - 2*q^191 + ...
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MATHEMATICA
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nmax = 100; CoefficientList[Series[Product[(1 - x^k)^2/(1 - x^(3*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* or *) nmax = 100; CoefficientList[Series[QPochhammer[x]^2 / QPochhammer[x^3], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 08 2020 *)
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PROG
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(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^3 + A), n))} /* Michael Somos, Jul 04 2012 */
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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