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Search: id:A116633
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| A116633 |
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Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts that are multiple of 3 (n>=0, 0<=k<=floor(n/3)). |
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3
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| 1, 1, 2, 2, 1, 4, 1, 5, 2, 7, 3, 1, 9, 5, 1, 13, 7, 2, 16, 10, 3, 1, 22, 14, 5, 1, 27, 20, 7, 2, 36, 26, 11, 3, 1, 44, 36, 15, 5, 1, 57, 47, 22, 7, 2, 70, 62, 29, 11, 3, 1, 89, 80, 41, 15, 5, 1, 108, 104, 54, 22, 7, 2, 135, 132, 73, 30, 11, 3, 1, 163, 169, 95, 42, 15, 5, 1, 202, 212
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row n has 1+floor(n/3) terms. Row sums yield the partition numbers (A000041). T(n,0)=A000726(n). T(n,1)=A116634(n). Sum(k*T(n,k),k=0..floor(n/3))=A116635(n).
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FORMULA
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G.f.=G(t,x)=1/product((1-x^(3j-2))(1-x^(3j-1))(1-tx^(3j)), j=1..infinity).
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EXAMPLE
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T(9,2)=3 because we have [6,3], [3,3,2,1], and [3,3,1,1,1].
Triangle starts:
1;
1;
2;
2,1;
4,1;
5,2;
7,3,1;
9,5,1;
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MAPLE
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g:=1/product((1-x^(3*j-2))*(1-x^(3*j-1))*(1-t*x^(3*j)), j=1..20): gser:=simplify(series(g, x=0, 26)): P[0]:=1: for n from 1 to 21 do P[n]:=coeff(gser, x^n) od: for n from 1 to 21 do seq(coeff(P[n], t, j), j=0..floor(n/3)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000041, A000726, A116634, A116635.
Sequence in context: A129721 A054995 A018219 this_sequence A134666 A129712 A051720
Adjacent sequences: A116630 A116631 A116632 this_sequence A116634 A116635 A116636
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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 19 2006
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