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Search: id:A116598
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| A116598 |
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Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n). |
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1
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| 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 2, 2, 1, 1, 0, 1, 4, 4, 2, 2, 1, 1, 0, 1, 7, 4, 4, 2, 2, 1, 1, 0, 1, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1, 24, 21, 14, 12, 8, 7, 4, 4, 2, 2
(list; table; graph; listen)
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OFFSET
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0,11
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COMMENT
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Row sums yield the partition numbers (A000041). T(n,0)=A002865(n), Sum(k*T(n,k),k=0..n)=A000070(n-1) for n>=1. Column k has g.f. x^k/product(1-x^j,j=2..infinity) (k>=0).
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FORMULA
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G.f.=G(t,x)=1/[(1-tx)*product(1-x^j, j=2..infinity)]. T(n,k)=p(n-k)-p(n-k-1) for k<n, where p(n) are the partition numbers (A000041).
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EXAMPLE
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T(6,2)=2 because we have [4,1,1] and [2,2,1,1].
Triangle starts:
1;
0,1;
1,0,1;
1,1,0,1;
2,1,1,0,1;
2,2,1,1,0,1;
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MAPLE
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with(combinat): T:=proc(n, k) if k<n then numbpart(n-k)-numbpart(n-k-1) elif k=n then 1 else 0 fi end: for n from 0 to 14 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, A002865, A000070.
Sequence in context: A029400 A069713 A072233 this_sequence A068914 A090824 A099314
Adjacent sequences: A116595 A116596 A116597 this_sequence A116599 A116600 A116601
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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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