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A101230
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Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.
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2
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1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 343, 484, 676, 935, 1282, 1744, 2355, 3158, 4208, 5573, 7340, 9616, 12536, 16266, 21012, 27028, 34628, 44196, 56204, 71226, 89964, 113270, 142180, 177948, 222089, 276430, 343172, 424959, 524966
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OFFSET
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0,2
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COMMENTS
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Note that if a partition of n has odd parts occur with even multiplicities then n must be even. This is the reason for only looking at partitions of 2n. - Michael Somos, Mar 04 2012
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LINKS
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FORMULA
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G.f.: product_{k>0}(1+x^k)/((1-x^k)(1+x^(3k)))= Theta_4(0, x^3)/theta(0, x)1/product_{k>0}(1-x^(3k)).
Euler transform of period 6 sequence [2, 1, 1, 1, 2, 1, ...]. - Vladeta Jovovic, Dec 17 2004
Expansion of q^(1/8) * eta(q^2) * eta(q^3) / (eta(q)^2 * eta(q^6)) in powers of q. - Michael Somos, Mar 04 2012
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EXAMPLE
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a(8)=12 because 8 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+2 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1.
1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + 113*x^9 + ...
1/q + 2*q^7 + 4*q^15 + 7*q^23 + 12*q^31 + 20*q^39 + 32*q^47 + 50*q^55 + 76*q^63 + ...
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MAPLE
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series(product((1+x^k)/((1-x^k)*(1+x^(3*k))), k=1..100), x=0, 100);
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1))*(1+x^(3*k-2)) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)
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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^2 + A) * eta(x^3 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Mar 04 2012 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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