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Search: id:A116599
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| A116599 |
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Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 2 (n>=0, 0<=k<=floor(n/2)). |
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1
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| 1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 2, 1, 6, 3, 1, 1, 8, 4, 2, 1, 11, 6, 3, 1, 1, 15, 8, 4, 2, 1, 20, 11, 6, 3, 1, 1, 26, 15, 8, 4, 2, 1, 35, 20, 11, 6, 3, 1, 1, 45, 26, 15, 8, 4, 2, 1, 58, 35, 20, 11, 6, 3, 1, 1, 75, 45, 26, 15, 8, 4, 2, 1, 96, 58, 35, 20, 11, 6, 3, 1, 1, 121, 75, 45, 26, 15, 8, 4, 2, 1
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the partition numbers (A000041). T(n,0)=A027336(n), Sum(k*T(n,k),k=0..floor(n/2))=A024786(n). Column k has g.f. x^(2k)/[(1-x)product(1-x^j,j=3..infinity)] (k=0,1,2,...).
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FORMULA
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G.f.=1/[(1-x)(1-tx^2)product(1-x^j, j=3..infinity)]. T(n,k)=p(n-2k)-p(n-2k-2) for k<=(n-2)/2; T(n, floor(n/2))=1 (follows at once from the g.f.).
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EXAMPLE
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T(6,1)=3 because we have [4,2], [3,2,1], and [2,1,1,1,1].
Triangle starts:
1;
1;
1,1;
2,1;
3,1,1;
4,2,1;
6,3,1,1;
8,4,2,1;
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MAPLE
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with(combinat): T:=proc(n, k) if k=floor(n/2) then 1 elif k<=(n-2)/2 then numbpart(n-2*k)-numbpart(n-2*k-2) fi end: for n from 0 to 18 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000041, A027336, A024786.
Sequence in context: A010766 A135841 A089178 this_sequence A138121 A138151 A122610
Adjacent sequences: A116596 A116597 A116598 this_sequence A116600 A116601 A116602
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 18 2006
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