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A116675
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Triangle read by rows: T(n,k) is the number of partitions of n into distinct part and having exactly k odd parts (n>=0, k>=0).
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2
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1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 3, 0, 2, 0, 2, 0, 5, 0, 2, 0, 4, 0, 7, 0, 1, 3, 0, 7, 0, 0, 10, 0, 2, 4, 0, 11, 0, 0, 14, 0, 4, 5, 0, 17, 0, 0, 19, 0, 8, 6, 0, 25, 0, 1, 0, 25, 0, 13, 0, 8, 0, 36, 0, 2, 0, 33, 0, 21, 0, 10, 0, 50, 0, 4, 0, 43, 0, 33, 0, 12, 0, 69, 0, 8, 0, 55, 0, 49, 0, 15, 0, 93, 0, 14
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Row n contains 1+floor(sqrt(n)) terms (at the end of certain rows there is an extra 0). Row sums yield A000009. T(n,0)=A035457(n) (n>=1); T(2n,0)=A000009(n), T(2n-1,0)=0. T(2n,1)=0, T(2n-1,1)=A036469(n). Sum(k*T(n,k), k>=0)=A116676(n).
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FORMULA
| G.f.=product((1+tx^(2j-1))(1+x^(2j)), j=1..infinity).
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EXAMPLE
| T(8,2)=4 because we have [7,1],[5,3],[5,2,1] and [4,3,1] ([8] and [6,2] do not qualify).
Triangle starts:
1;
0,1;
1,0;
0,2;
1,0,1;
0,3,0;
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MAPLE
| g:=product((1+t*x^(2*j-1))*(1+x^(2*j)), j=1..25): gser:=simplify(series(g, x=0, 38)): P[0]:=1: for n from 1 to 26 do P[n]:=sort(coeff(gser, x^n)) od: for n from 0 to 26 do seq(coeff(P[n], t, j), j=0..floor(sqrt(n))) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A000009, A035457, A036469, A116676.
Sequence in context: A158461 A118269 A144152 * A123022 A072943 A072175
Adjacent sequences: A116672 A116673 A116674 * A116676 A116677 A116678
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 22 2006
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