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Search: id:A116663
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| A116663 |
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Triangle read by rows: T(n,k) = number of partitions of n into odd parts and having exactly k parts equal to 1 (n>=0, 1<=k<=n). |
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+0
1
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| 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 3, 3, 2, 2, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1
(list; table; graph; listen)
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OFFSET
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0,46
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COMMENT
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Row sums yield A000009. T(n,0)=A087897(n). Column k has g.f.=x^k/Product(1-x^(2j-1), j=2..infinity) (all columns are basically identical). Sum(k*T(n,k),k=0..n)=A036469(n).
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FORMULA
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G.f.=1/[(1-tx)*Product(1-x^(2j-1), j=2..infinity)].
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EXAMPLE
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T(10,1)=2 because the only partitions of 10 into odd parts and having exactly 1 part equal to 1 are [9,1] and [3,3,3,1].
Triangle starts:
1;
0,1;
0,0,1;
1,0,0,1;
0,1,0,0,1;
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MAPLE
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g:=1/(1-t*x)/product(1-x^(2*j-1), j=2..30): gser:=simplify(series(g, x=0, 18)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(gser, x^n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000009, A087897, A036469.
Sequence in context: A076882 A016397 A037908 this_sequence A074871 A050372 A037802
Adjacent sequences: A116660 A116661 A116662 this_sequence A116664 A116665 A116666
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2006
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