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A079499
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Total number of parts in all partitions of n into distinct odd parts.
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5
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0, 1, 0, 1, 2, 1, 2, 1, 4, 4, 4, 4, 6, 7, 6, 10, 12, 13, 12, 16, 18, 22, 22, 25, 32, 36, 36, 42, 50, 53, 58, 64, 76, 83, 88, 99, 116, 123, 132, 147, 168, 181, 194, 215, 240, 262, 280, 306, 346, 375, 396
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OFFSET
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0,5
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COMMENTS
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Also sum of the sizes of the Durfee squares of all self-conjugate partitions of n. Example: a(13)=7 because there are three self-conjugate partitions of 13, namely [7,1,1,1,1,1,1], [5,3,3,1,1] and [4,4,3,2], having Durfee squares of sizes 1,3 and 3, respectively. a(n)=sum(k*A116422(n,k),k=1..floor(sqrt(n))). - Emeric Deutsch, Feb 14 2006
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REFERENCES
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G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).
A. Knopfmacher and N. Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
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FORMULA
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G.f.: sum x^(2k-1)/(1+x^(2k-1); k=1..inf * prod (1+x^(2m-1); m=1..inf
Sum_{k>0} (k*x^(k^2)/Product_{j=1..k} (1-x^(2*j))). - Vladeta Jovovic, Aug 06 2004
G.f.=sum(kx^(k^2)/product(1-x^(2i),i =1..k),k=1..infinity). - Emeric Deutsch, Feb 14 2006
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EXAMPLE
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a(13)=7 because the partitions of 13 into distinct odd parts are [13], [9,3,1] and [7,5,1] and we have 1+3+3=7 parts.
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MAPLE
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g:=sum(k*x^(k^2)/product(1-x^(2*i), i =1..k), k=1..20):gser:=series(g, x=0, 52): seq(coeff(gser, x, n), n=0..50); - Emeric Deutsch, Feb 14 2006
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CROSSREFS
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Cf. A015723, A000700, A067619, A006128.
Cf. A032021.
Cf. A116422.
Sequence in context: A188440 A216327 A099875 * A166235 A143591 A085063
Adjacent sequences: A079496 A079497 A079498 * A079500 A079501 A079502
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KEYWORD
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nonn,changed
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AUTHOR
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Arnold Knopfmacher (arnoldk(AT)cam.wits.ac.za), Jan 21 2003
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STATUS
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approved
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