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Search: id:A101198
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| A101198 |
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Number of partitions of n with rank 1 (the rank of a partition is the largest part minus the number of parts). |
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+0
6
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| 0, 1, 0, 1, 1, 2, 1, 3, 3, 5, 5, 8, 8, 13, 14, 20, 23, 31, 35, 48, 55, 72, 84, 108, 126, 160, 187, 233, 275, 340, 398, 489, 574, 697, 819, 988, 1158, 1390, 1627, 1941, 2271, 2696, 3145, 3721, 4335, 5104, 5938, 6967, 8088, 9462, 10964, 12783
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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Column k=1 in the triangle A063995.
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REFERENCES
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George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
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FORMULA
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G.f. for the number of partitions of n with rank r is Sum((-1)^k*x^(r*k)*(x^((3*k^2+k)/2)-x^((3*k^2-k)/2)), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 20 2004
Also Sum(x^(2*n+r+1)*Product((1-x^(2*n+r+1-k))/(1-x^k),k=1..n),n=0..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 05 2008
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EXAMPLE
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a(6)=2 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111
have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.
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MAPLE
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with(combinat): for n from 1 to 35 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=1 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..35);
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CROSSREFS
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Cf. A000041, A063995.
Cf. A101198-A101200, A101707-A101709.
Adjacent sequences: A101195 A101196 A101197 this_sequence A101199 A101200 A101201
Sequence in context: A074500 A107237 A070047 this_sequence A034394 A058689 A059876
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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), Dec 12 2004
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