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A092319
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Sum of smallest parts of all partitions of n into odd distinct parts.
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4
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1, 0, 3, 1, 5, 1, 7, 4, 10, 4, 12, 9, 15, 9, 20, 17, 23, 17, 28, 27, 36, 28, 41, 43, 50, 44, 62, 62, 71, 66, 84, 91, 103, 96, 119, 127, 139, 137, 167, 178, 191, 192, 223, 241, 266, 264, 302, 331, 351, 360, 411, 439, 469, 485, 542, 587, 628, 646, 714, 773, 819, 854, 945
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: Sum((2*n-1)*x^(2*n-1)*Product(1+x^(2*k+1), k = n .. infinity), n = 1 .. infinity).
a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/6)) / (2^(7/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018
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EXAMPLE
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a(13)=15 because the partitions of 13 into distinct odd parts are [13],[9,3,1] and [7,5,1], with sum of the smallest terms 13+1+1=15.
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MAPLE
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f:=sum((2*n-1)*x^(2*n-1)*product(1+x^(2*k+1), k=n..40), n=1..40): fser:=simplify(series(f, x=0, 66)): seq(coeff(fser, x^n), n=1..63); # Emeric Deutsch, Feb 27 2006
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, b(n, i+2)+b(n-i, i+2)))
end:
a:= n-> add(`if`(j::odd, j*b(n-j, j+2), 0), j=1..n):
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MATHEMATICA
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nmax = 60; Rest[CoefficientList[Series[Sum[(2*k - 1)*x^(2*k - 1) * Product[1 + x^(2*j + 1), {j, k, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 28 2016 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004
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STATUS
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approved
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