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Search: id:A117470
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| A117470 |
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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 occurs and the difference between the largest part and the smallest part is k (n>=0, k>=0). |
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+0
2
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| 1, 2, 2, 1, 3, 1, 2, 3, 4, 2, 1, 2, 5, 1, 4, 4, 2, 3, 6, 4, 4, 6, 4, 1, 2, 9, 6, 1, 6, 6, 9, 2, 2, 11, 10, 3, 4, 10, 11, 6, 4, 11, 17, 6, 1, 5, 11, 17, 10, 1, 2, 15, 21, 12, 2, 6, 12, 24, 18, 3, 2, 17, 28, 20, 5, 6, 14, 31, 26, 8, 4, 17, 38, 31, 10, 1, 4, 18, 37, 41, 14, 1, 2, 21, 45, 45, 19, 2, 8
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Row n contains ceil((sqrt(9+8n)-3)/2) terms, i.e. 1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,... . Row sums yield A034296. T(n,0)=A000005(n) (number of divisors of n). Sum(k*T(n,k),k>=0)=A117471(n).
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FORMULA
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G.f.=G(t,x)=sum(x^j*product(1+tx^i, i=1..j-1)/(1-x^j), j=1..infinity).
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EXAMPLE
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T(9,2)=4 because we have [4,3,2],[3,3,2,1],[3,2,2,1,1], and [3,2,1,1,1,1].
Triangle starts:
1;
2;
2,1;
3,1;
2,3;
4,2,1;
2,5,1;
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MAPLE
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g:=sum(x^j*product(1+t*x^i, i=1..j-1)/(1-x^j), j=1..30): gser:=simplify(series(g, x=0, 28)): for n from 1 to 28 do P[n]:=sort(coeff(gser, x^n)) od: for n from 1 to 25 do seq(coeff(P[n], t, j), j=0..ceil((sqrt(9+8*n)-5)/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A034296, A000005, A117471.
Sequence in context: A051135 A135352 A072528 this_sequence A070786 A117500 A044924
Adjacent sequences: A117467 A117468 A117469 this_sequence A117471 A117472 A117473
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 20 2006
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