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A066897
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Total number of odd parts in all partitions of n.
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13
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1, 2, 5, 8, 15, 24, 39, 58, 90, 130, 190, 268, 379, 522, 722, 974, 1317, 1754, 2330, 3058, 4010, 5200, 6731, 8642, 11068, 14076, 17864, 22528, 28347, 35490, 44320, 55100, 68355, 84450, 104111, 127898, 156779, 191574, 233625, 284070, 344745
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Contribution from Omar E. Pol, Feb 12 2012 (Start):
It appears that a(n) is also the sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n, with their parts written in nonincreasing order.
Example: a(4) = 12 - 5 + 2 - 1 = 14 - 6 = 8.
The calculation is
. 4 = 4
. 3 - 1 = 2
. 2 - 2 = 0
. 2 - 1 + 1 = 2
. 1 - 1 + 1 - 1 = 0
----------------------
. 12 - 5 + 2 - 1 = 8
(End)
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FORMULA
| Sum_{k=1..n} b(k)*numbpart(n-k), where b(k)=A001227(k)=number of odd divisors of k. - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 26 2002
a(n)=sum(k*A103919(n,k),k=0..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2006
G.f.=sum(x^(2j-1)/(1-x^(2j-1)), j=1..infinity)/product(1-x^j, j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2006
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EXAMPLE
| a(4)=8 because in the partitions of 4, namely [4],[3,1],[2,2],[2,1,1],[1,1,1,1], we have a total of 0+2+0+2+4=8 odd parts.
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MAPLE
| g:=sum(x^(2*j-1)/(1-x^(2*j-1)), j=1..70)/product(1-x^j, j=1..70): gser:=series(g, x=0, 45): seq(coeff(gser, x^n), n=1..44); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 13 2006
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CROSSREFS
| Cf. A000041.
Cf. A001227, A006128, A066898.
Cf. A103919.
Sequence in context: A121641 A058884 A073335 * A078697 A066629 A154327
Adjacent sequences: A066894 A066895 A066896 * A066898 A066899 A066900
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Naohiro Nomoto (n_nomoto(AT)yabumi.com), Jan 24 2002
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 26 2002
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