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A238210
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The total number of 3's in all partitions of n into an odd number of distinct parts.
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2
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0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 23, 28, 32, 37, 44, 51, 58, 68, 78, 89, 103, 118, 134, 154, 175, 199, 227, 257, 291, 330, 373, 421, 475, 535, 602, 677, 760, 852, 955, 1069, 1196, 1336, 1491, 1663, 1853, 2063, 2296
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OFFSET
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0,10
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COMMENTS
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The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).
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LINKS
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FORMULA
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a(n) = Sum_{j=1..round(n/6)} A067661(n-(2*j-1)*3) - Sum_{j=1..floor(n/6)} A067659(n-6*j).
G.f.: (1/2)*(x^3/(1+x^3))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^3/(1-x^3))*(Product_{n>=1} 1 - x^n).
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EXAMPLE
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a(12) = 3 because the partitions in question are: 8+3+1, 7+3+2, 5+4+3.
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PROG
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(PARI) seq(n)={my(A=O(x^(n-2))); Vec(x^3*(eta(x^2 + A)/(eta(x + A)*(1+x^3)) + eta(x + A)/(1-x^3))/2, -(n+1))} \\ Andrew Howroyd, May 01 2020
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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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