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Search: id:A090858
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| A090858 |
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Number of partitions of n such that there is only one part which occurs twice, while all other parts occur only once. |
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+0
2
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| 0, 1, 0, 2, 2, 2, 4, 6, 7, 8, 13, 15, 21, 25, 30, 39, 50, 58, 74, 89, 105, 129, 156, 185, 221, 264, 309, 366, 433, 505, 593, 696, 805, 941, 1090, 1258, 1458, 1684, 1933, 2225, 2555, 2922, 3346, 3823, 4349, 4961, 5644, 6402, 7267, 8234, 9309, 10525, 11886, 13393
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Number of solutions (p(1),p(2),...,p(n)), p(i)>=0,i=1..n, to p(1)+2*p(2)+...+n*p(n)=n such that |{i: p(i)<>0}| = p(1)+p(2)+...+p(n)-1.
Also number of partitions of n such that if k is the largest part, then, with exactly one exception, all the integers 1,2,...,k occur as parts. Example: a(7)=4 because we have [4,2,1],[3,3,1],[3,2,2], and[3,1,1,1,1]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
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FORMULA
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G.f.: Sum_{k>0} x^(2*k)/(1+x^k) * Product_{k>0} (1+x^k). Convolution of 1-A048272(n) and A000009(n). a(n) = A036469(n) - A015723(n).
G.f.=sum(x^(k(k+1)/2)[(1-x^k)/x^(k-1)/(1-x)-k]/product(1-x^i,i=1..k), k=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
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EXAMPLE
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a(7)= 4 because we have 4 such partitions of 7: [1,1,2,3], [1,1,5], [2,2,3], [1,3,3].
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MAPLE
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g:=sum(x^(k*(k+1)/2)*((1-x^k)/x^(k-1)/(1-x)-k)/product(1-x^i, i=1..k), k=1..15): gser:=series(g, x=0, 64): seq(coeff(gser, x, n), n=1..54); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 18 2006
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CROSSREFS
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Cf. A047967.
Adjacent sequences: A090855 A090856 A090857 this_sequence A090859 A090860 A090861
Sequence in context: A096575 A002722 A093393 this_sequence A036654 A010558 A060827
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KEYWORD
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easy,nonn
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AUTHOR
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Vladeta Jovovic (vladeta(AT)Eunet.yu), Feb 12 2004
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004
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