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A113685
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Triangular array T(n,k)=number of partitions of n in which sum of odd parts is k, for k=0,1,...n; n>=0.
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6
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1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 3, 3, 0, 2, 0, 2, 0, 4, 0, 3, 0, 4, 0, 3, 0, 5, 5, 0, 3, 0, 4, 0, 4, 0, 6, 0, 5, 0, 6, 0, 6, 0, 5, 0, 8, 7, 0, 5, 0, 6, 0, 8, 0, 6, 0, 10, 0, 7, 0, 10, 0, 9, 0, 10, 0, 8, 0, 12, 11, 0, 7, 0, 10, 0, 12, 0, 12, 0, 10, 0, 15, 0, 11, 0, 14, 0, 15, 0
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,10
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COMMENTS
| (Sum over row n) = A000041(n) = number of partitions of n. Reversal of this array is array in A113686, except for row 0.
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 11 2010: (Start)
Let M = an infinite lower triangular matrix with A000041 interleaved with zeros:
(1, 0, 1, 0, 2, 0, 3, 0, 5,...) and Q = A000009 diagonalized with the rest
zeros. Then A113685 = M*Q. That row sums of the triangle (deleting prefaced
zeros) = A000041 is equivalent to the identity: p(x) = p(x^2) * A000009(x). (End)
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FORMULA
| G.f.=G(t,x)=1/product((1-t^(2j-1)x^(2j-1))(1-x^(2j)), j=1..infinity). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
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EXAMPLE
| First 5 rows:
0
0 1
1 0 1
0 1 0 2
2 0 1 0 2
0 2 0 2 0 3.
The partitions of 5 are
5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1;
sums of odd parts are 5,1,3,5,1,3,5, respectively,
so that the numbers of 0's, 1's, 2s, 3s, 4s, 5s
are 0,2,0,2,0,3, which is row 5 of the array.
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MAPLE
| g:=1/product((1-t^(2*j-1)*x^(2*j-1))*(1-x^(2*j)), j=1..20): gser:=simplify(series(g, x=0, 22)): P[0]:=1: for n from 1 to 14 do P[n]:=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 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
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CROSSREFS
| Cf. A000041, A113686.
Cf. A066967.
Sequence in context: A049502 A204423 A112170 * A049825 A039651 A038190
Adjacent sequences: A113682 A113683 A113684 * A113686 A113687 A113688
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu), Nov 05 2005
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 17 2006
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