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A113685
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Triangular array read by rows: T(n,k) is the 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
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OFFSET
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0,10
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COMMENTS
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(Sum over row n) = A000041(n) = number of partitions of n.
Reversal of this array is array in A113686.
From Gary W. Adamson, 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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LINKS
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Table of n, a(n) for n=0..97.
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FORMULA
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G.f.: G(t,x) = 1/Product_{j>=1} (1 - t^(2j-1)*x^(2j-1))*(1-x^(2j)). - Emeric Deutsch, Feb 17 2006
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EXAMPLE
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First 5 rows:
1;
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.
The 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
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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, Feb 17 2006
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CROSSREFS
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Cf. A000041, A113686, A066967.
Sequence in context: A204423 A112170 A259976 * A049825 A287443 A039651
Adjacent sequences: A113682 A113683 A113684 * A113686 A113687 A113688
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling, Nov 05 2005
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EXTENSIONS
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More terms from Emeric Deutsch, Feb 17 2006
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STATUS
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approved
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