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Search: id:A077285
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| A077285 |
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Number of partitions of n with designated summands. |
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+0
6
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| 1, 1, 3, 5, 10, 15, 28, 41, 69, 102, 160, 231, 352, 498, 732, 1027, 1470, 2031, 2856, 3896, 5382, 7272, 9896, 13233, 17800, 23579, 31362, 41219, 54288, 70791, 92456, 119698, 155097, 199512, 256664, 328134, 419436, 533162, 677412, 856573
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Expansion of eta(q^6)/(eta(q)eta(q^2)eta(q^3)) in powers of q. - Michael Somos Feb 05 2004
Euler transform of period 6 sequence [1,2,2,2,1,2,...]. - Michael Somos Feb 05 2004
Sum of products of multiplicities of parts in all partitions of n. The partitions of 4 are 4, 1+3, 2+2, 2+1+1, 1+1+1+1, the corresponding products are 1,1,2,2,4, and their sum is a(4) = 10. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 16 2005
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REFERENCES
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Andrews, G. E.; Lewis, R. P.; Lovejoy, J. Partitions with designated summands. Acta Arith. 105 (2002), no. 1, 51-66.
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LINKS
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N. J. A. Sloane, Transforms
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FORMULA
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Generating function is P(x)P(x^2)P(x^3)/P(x^6), where P(x)=Product_{k>0} 1/(1-x^k) is the partition generating function (A000041)
EULER(DCONV(A000012, iEULER(A000027)))
G.f.: Product((1-x^i+x^(2*i))/(1-x^i)^2, i = 1 .. infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jan 16 2005
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EXAMPLE
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a(3)=5 because the partitions of 3 with designated summands are 3', 2'1', 1'11, 11'1, 111'
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PROGRAM
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(PARI) a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(X^6)/(eta(X)*eta(X^2)*eta(X^3)), n))
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CROSSREFS
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Cf. A000041, A091601.
Adjacent sequences: A077282 A077283 A077284 this_sequence A077286 A077287 A077288
Sequence in context: A097513 A045513 A008337 this_sequence A072523 A054473 A006168
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KEYWORD
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nonn
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AUTHOR
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Jorn B. Olsson (olsson(AT)math.ku.dk), Nov 26 2003
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EXTENSIONS
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Edited and extended by Christian G. Bower (bowerc(AT)usa.net), Jan 23 2004
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