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A117466
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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 k to the largest part occurs (1<=k<=n).
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2
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1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 3, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 5, 1, 1, 0, 0, 0, 1, 6, 2, 0, 1, 0, 0, 0, 1, 8, 2, 1, 1, 0, 0, 0, 0, 1, 10, 2, 1, 0, 1, 0, 0, 0, 0, 1, 12, 3, 1, 0, 1, 0, 0, 0, 0, 0, 1, 15, 3, 2, 1, 0, 1, 0, 0, 0, 0, 0, 1, 18, 4, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 22, 5, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| Row sums yield A034296. T(n,1)=A000009(n). sum(k*T(n,k),k=1..n)=A117467(n).
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FORMULA
| G.f.=G(t,x)=sum(tx^j*product(1+x^i, i=1..j-1)/(1-tx^j), j=1..infinity).
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EXAMPLE
| T(11,2)=3 because we have [4,3,2,2],[3,3,3,2] and [3,2,2,2,2].
Triangle starts:
1;
1,1;
2,0,1;
2,1,0,1;
3,1,0,0,1;
4,1,1,0,0,1;
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MAPLE
| g:=sum(t*x^j*product(1+x^i, i=1..j-1)/(1-t*x^j), j=1..50): gser:=simplify(series(g, x=0, 17)): for n from 1 to 14 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 14 do seq(coeff(P[n], t, j), j=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A034296, A000009, A117467.
Sequence in context: A140224 A075993 A117170 * A136266 A054523 A161363
Adjacent sequences: A117463 A117464 A117465 * A117467 A117468 A117469
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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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