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A238131
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Number of parts in all partitions of n into odd number of distinct parts.
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5
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0, 1, 1, 1, 1, 1, 4, 4, 7, 10, 13, 16, 22, 25, 31, 42, 48, 59, 73, 89, 108, 132, 156, 190, 227, 271, 318, 380, 449, 526, 618, 722, 841, 980, 1138, 1321, 1526, 1760, 2028, 2333, 2683, 3070, 3517, 4017, 4584, 5228, 5948, 6757, 7673, 8696, 9845, 11132, 12577
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OFFSET
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0,7
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LINKS
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FORMULA
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G.f.: (1/2)*prod(k>=1, 1+x^k ) * sum(k>=1, x^k/(1+x^k) ) + (1/2)*prod(k>=1, 1-x^k) * sum(k>=1, x^k/(1-x^k) ).
G.f.: (2 * (x; x)_inf * (log(1-x) + psi_x(1)) - (-1; x)_inf * (log(1-x) + psi_x(1-log(-1)/log(x))))/(4*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 27 2018
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EXAMPLE
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a(8)=7 because the partitions of 8 into odd number of distinct parts are: 8, 5+2+1 and 4+3+1.
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MAPLE
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b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, [1, 0$3], b(n, i-1)+`if`(i>n, 0, (p->
[p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
end:
a:= n-> b(n$2)[4]:
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MATHEMATICA
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max = 50; s = (1/2)*Product[1+x^k, {k, 1, max}]*Sum[x^k/(1+x^k), {k, 1, max}] + (1/2)*Product[1-x^k, {k, 1, max}]*Sum[x^k/(1-x^k), {k, 1, max}] + O[x]^(max+1); CoefficientList[s, x] (* Jean-François Alcover, Dec 27 2015 *)
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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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