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Search: id:A116674
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| A116674 |
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Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having exactly k distinct parts (n>=1, k>=1). |
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+0
2
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| 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 3, 1, 5, 3, 4, 1, 2, 7, 1, 2, 8, 2, 2, 10, 3, 2, 11, 5, 2, 13, 7, 4, 12, 11, 1, 19, 11, 1, 2, 18, 17, 1, 3, 20, 21, 2, 2, 22, 27, 3, 2, 25, 32, 5, 4, 24, 41, 7, 2, 30, 46, 11, 2, 31, 56, 15, 2, 36, 62, 22, 3, 33, 80, 25, 1, 2, 39, 87, 36, 1, 4, 38, 103, 45, 2, 2, 45
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row n has floor(sqrt(n)) terms. Row sums yield A000009. T(n,1)=A001227(n) (n>=1). Sum(k*T(n,k),k>=1)=A038348(n-1) (n>=1).
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FORMULA
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G.f.=product(1+tx^(2j-1)/(1-x^(2j-1)), j=1..infinity).
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EXAMPLE
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T(9,2)=4 because the only partitions of 9 into odd parts and having 2 distinct parts are [7,1,1],[5,1,1,1,1],[3,3,1,1,1], and [3,1,1,1,1,1,1].
Triangle starts:
1;
1;
2;
1,1;
2,1;
2,2;
2,3;
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MAPLE
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g:=product(1+t*x^(2*j-1)/(1-x^(2*j-1)), j=1..35): gser:=simplify(series(g, x=0, 34)): for n from 1 to 29 do P[n]:=coeff(gser, x^n) od: for n from 1 to 29 do seq(coeff(P[n], t, j), j=1..floor(sqrt(n))) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000009, A001227, A038348.
Adjacent sequences: A116671 A116672 A116673 this_sequence A116675 A116676 A116677
Sequence in context: A141272 A124038 A029311 this_sequence A025836 A029319 A050205
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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), Feb 22 2006
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