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A238215
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The total number of 1's in all partitions of n into an even number of distinct parts.
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2
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0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 13, 15, 18, 21, 24, 28, 33, 38, 44, 51, 59, 68, 79, 90, 104, 119, 136, 156, 178, 202, 230, 261, 296, 335, 379, 427, 482, 543, 610, 686, 770, 863, 967, 1082, 1209, 1351, 1508, 1681, 1873, 2085, 2318, 2577
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OFFSET
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0,11
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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/2)} A067659(n-(2*j-1)) - Sum_{j=1..floor(n/2)} A067661(n-2*j).
G.f.: (1/2)*(x/(1+x))*(Product_{n>=1} 1 + x^n) - (1/2)*(x/(1-x))*Product_{n>=1} 1 - x^n).
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EXAMPLE
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a(12) = 3 because the partitions in question are: 11+1, 6+3+2+1, 5+4+2+1.
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0, t,
`if`(i>n, 0, b(n, i+1, t)+b(n-i, i+1, 1-t)))
end:
a:= n-> b(n-1, 2, 0):
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MATHEMATICA
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b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i + 1, 1 - t]]];
a[n_] := b[n - 1, 2, 0];
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PROG
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(PARI) seq(n)={my(A=O(x^n)); Vec(x*(eta(x^2 + A)/(eta(x + A)*(1+x)) - eta(x + A)/(1-x))/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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