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Search: id:A117995
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| A117995 |
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Number of partitions of n in which both smallest and largest part occur only once. |
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+0
2
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| 0, 0, 1, 1, 2, 3, 4, 6, 8, 11, 14, 20, 24, 33, 41, 54, 66, 87, 105, 136, 165, 209, 253, 319, 383, 477, 574, 707, 847, 1038, 1238, 1506, 1794, 2166, 2573, 3093, 3660, 4377, 5170, 6152, 7245, 8590, 10087, 11913, 13959, 16423, 19196, 22518, 26252, 30700, 35717
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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Also number of partitions of n in which the least part is 1 and if k is the largest part, then k>=2 and k-1 also occurs. Example: a(8)=6 because we have [4,3,1],[3,2,2,1],[3,2,1,1,1],[2,2,2,1,1],[2,2,1,1,1,1] and [2,1,1,1,1,1,1].
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FORMULA
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G.f.=sum(sum(x^(j+k)/product(1-x^i, i=j+1..k-1), j=1..k-1), k=2..infinity). G.f.=x^3/[(1-x)(1-x^2)]+sum(x^(2k)/product(1-x^j, j=1..k), k=3..infinity).
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EXAMPLE
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a(8)=6 because we have [7,1],[6,2],[5,3],[5,2,1],[4,3,1] and [3,2,2,1].
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MAPLE
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g:=x^3/(1-x)/(1-x^2)+sum(x^(2*k)/product(1-x^j, j=1..k), k=3..70): gser:=series(g, x=0, 60): seq(coeff(gser, x, n), n=1..55);
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CROSSREFS
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Sequence in context: A038348 A035945 A094707 this_sequence A033834 A127419 A132217
Adjacent sequences: A117992 A117993 A117994 this_sequence A117996 A117997 A117998
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 08 2006
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