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A117989
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Number of partitions of n such that the least part occurs at least twice.
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3
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0, 1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 67, 94, 121, 165, 209, 280, 353, 462, 582, 749, 935, 1192, 1480, 1862, 2302, 2871, 3526, 4366, 5335, 6555, 7976, 9737, 11789, 14317, 17259, 20845, 25032, 30093, 35992, 43087, 51347, 61216, 72710, 86362, 102235
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| More generally, g.f. for number of partitions of n such that the least part occurs at least m times is sum(x^(mk)/product(1-x^j, j=k..infinity), k=1..infinity). Also number of partitions of n such that if k is the largest part, then k>=2 and k-1 does not occur. Example: a(5)=3 because we have [5],[4,1] and [3,1,1].
Also number of partitions of 2*n such that the difference between greatest part and smallest part is n. - Vladeta Jovovic (vladeta(AT)eunet.rs), May 09 2008
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FORMULA
| G.f.=sum(x^(2k)/product(1-x^j, j=k..infinity), k=1..infinity). G.f.=sum(x^k*(1-x^(k-1))/product(1-x^j, j=1..k), k=2..infinity).
a(n) = 2*A000041(n)-A000041(n+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 21 2006
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EXAMPLE
| a(5)=3 because we have [3,1,1],[2,1,1,1] and [1,1,1,1,1].
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MAPLE
| g:=sum(x^k*(1-x^(k-1))/product(1-x^j, j=1..k), k=2..70): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=1..50);
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MATHEMATICA
| Table[Length[Select[IntegerPartitions[n], Count[#, Min[#]]>1&]], {n, 50}] (* From Harvey P. Dale, Apr 23 2011 *)
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CROSSREFS
| Cf. A096373.
Cf. A097364.
Sequence in context: A178238 A200792 A161416 * A086543 A110618 A108046
Adjacent sequences: A117986 A117987 A117988 * A117990 A117991 A117992
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 08 2006
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