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Search: id:A115604
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| A115604 |
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Triangle read by rows: T(n,k) is the number of partitions of n into odd parts in which the smallest part occurs k times (1<=k<=n). |
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+0
1
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| 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 2, 1, 0, 1, 0, 0, 1, 3, 2, 1, 1, 1, 0, 1, 0, 0, 1, 3, 3, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4, 2, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 5, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 5, 4, 3, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1
(list; table; graph; listen)
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OFFSET
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1,22
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COMMENT
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Row sums yield A000009. T(n,1)=A087897(n+2). Sum(k*T(n,k),k=1..n)=A092268(n).
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FORMULA
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G.f.=G(t,x)=sum(tx^(2k-1)/[(1-tx^(2k-1))product(1-x^(2i-1), i=k+1..infinity)], k=1..infinity).
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EXAMPLE
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T(14,2)=4 because we have [9,3,1,1],[7,7],[7,5,1,1], and [3,3,3,3,1,1].
Triangle starts:
1;
0,1;
1,0,1;
1,0,0,1;
1,1,0,0,1;
1,1,1,0,0,1;
2,1,0,1,0,0,1;
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MAPLE
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g:=sum(t*x^(2*k-1)/(1-t*x^(2*k-1))/product(1-x^(2*i-1), i=k+1..40), k=1..40): gser:=simplify(series(g, x=0, 55)): for n from 1 to 15 do P[n]:=expand(coeff(gser, x^n)) od: for n from 1 to 15 do seq(coeff(P[n], t^j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000009, A087897, A092268, A117408.
Sequence in context: A073423 A134023 A015738 this_sequence A128617 A116488 A145765
Adjacent sequences: A115601 A115602 A115603 this_sequence A115605 A115606 A115607
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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), Mar 13 2006
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