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Search: id:A118806
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| A118806 |
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Triangle read by rows: T(n,k) is the number of partitions of n having k parts of multiplicity 3 (n,k>=0). |
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+0
4
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| 1, 1, 2, 2, 1, 5, 6, 1, 9, 2, 12, 3, 19, 3, 24, 5, 1, 34, 8, 43, 13, 62, 13, 2, 77, 23, 1, 105, 28, 2, 132, 40, 4, 177, 49, 5, 220, 71, 6, 287, 89, 8, 1, 356, 123, 11, 462, 147, 18, 570, 198, 23, 1, 723, 249, 29, 1, 888, 329, 37, 1, 1121, 400, 50, 4, 1370, 518, 69, 1, 1705, 642, 85
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OFFSET
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0,3
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COMMENT
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T(n,0)=A118807(n). T(n,1)=A118808(n). Row sums yield the partition numbers (A000041). Sum(k*T(n,k), k>=0)=A117524(n) (n>=1).
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FORMULA
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G.f.=product(1+x^j+x^(2j)+tx^(3j)+x^(4j)/(1-x^j), j=1..infinity).
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EXAMPLE
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T(12,2)=2 because we have [3,3,3,1,1,1] and [3,2,2,2,1,1,1].
Triangle starts:
1;
1;
2;
2,1;
5;
6,1;
9,2;
12,3;
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MAPLE
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g:=product(1+x^j+x^(2*j)+t*x^(3*j)+x^(4*j)/(1-x^j), j=1..35): gser:=simplify(series(g, x=0, 35)): P[0]:=1: for n from 1 to 30 do P[n]:=coeff(gser, x^n) od: for n from 0 to 30 do seq(coeff(P[n], t, j), j=0..degree(P[n])) od; # sequence given in triangular form
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CROSSREFS
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Cf. A000041, A118807, A118808, A117524, A116595, A116644.
Sequence in context: A133611 A010094 A019710 this_sequence A124644 A056857 A129100
Adjacent sequences: A118803 A118804 A118805 this_sequence A118807 A118808 A118809
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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), Apr 29 2006
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