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A117468
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Triangle read by rows: T(n,k) is the number of partitions of n in which every integer from the smallest part to the largest part k occurs (1<=k<=n).
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1
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1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 0, 1, 1, 3, 2, 0, 0, 1, 1, 3, 2, 1, 0, 0, 1, 1, 4, 3, 1, 0, 0, 0, 1, 1, 4, 5, 1, 1, 0, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 0, 0, 1, 1, 5, 7, 3, 0, 1, 0, 0, 0, 0, 1, 1, 6, 9, 4, 1, 1, 0, 0, 0, 0, 0, 1, 1, 6, 10, 6, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 7, 12, 7, 2, 0, 1, 0, 0, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,8
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COMMENTS
| Also number of partitions of n having k parts and such that all parts smaller than the largest part occur only once. Row sums yield A034296. T(n,1)=T(n,n)=1 sum(k*T(n,k),k=1..n)=A117469(n).
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FORMULA
| G.f.=G(t,x)=sum(tx^j*product(1+tx^i, i=1..j-1)/(1-tx^j), j=1..infinity).
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EXAMPLE
| T(10,3)=5 because we have [3,3,2,2],[3,3,2,1,1],[3,2,2,2,1],[3,2,2,1,1,1] and [3,2,1,1,1,1,1].
Triangle starts:
1;
1,1;
1,1,1;
1,2,0,1;
1,2,1,0,1;
1,3,2,0,0,1;
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MAPLE
| g:=sum(t*x^j*product(1+t*x^i, i=1..j-1)/(1-t*x^j), j=1..50): gser:=simplify(series(g, x=0, 18)): for n from 1 to 15 do P[n]:=sort(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
| Cf. A034296, A117469.
Sequence in context: A155041 A108455 A193759 * A116374 A025911 A060184
Adjacent sequences: A117465 A117466 A117467 * A117469 A117470 A117471
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KEYWORD
| nonn,tabl
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 19 2006
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