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A054242
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Triangle read by rows: row n (n>=0) gives the number of partitions of (n,0), (n-1,1), (n-2,2), ..., (0,n) respectively into sums of distinct pairs.
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13
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1, 1, 1, 1, 2, 1, 2, 3, 3, 2, 2, 5, 5, 5, 2, 3, 7, 9, 9, 7, 3, 4, 10, 14, 17, 14, 10, 4, 5, 14, 21, 27, 27, 21, 14, 5, 6, 19, 31, 42, 46, 42, 31, 19, 6, 8, 25, 44, 64, 74, 74, 64, 44, 25, 8, 10, 33, 61, 93, 116, 123, 116, 93, 61, 33, 10
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OFFSET
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0,5
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COMMENTS
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By analogy with ordinary partitions into distinct parts (A000009). The empty partition gives T(0,0)=1 by definition. A054225 and A201376 give pair partitions with repeats allowed.
Also number of partitions into pairs which are not both even.
In the paper by S. M. Luthra: "Partitions of bipartite numbers when the summands are unequal", the square table on page 370 contains an errors. In the formula (6, p. 372) for fixed m there should be factor 1/m!. The correct asymptotic formula is q(m, n) ~ (sqrt(12*n)/Pi)^m * exp(Pi*sqrt(n/3)) / (4*3^(1/4)*m!*n^(3/4)). The same error is also in article by F. C. Auluck (see A054225). - Vaclav Kotesovec, Feb 02 2016
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LINKS
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FORMULA
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G.f.: (1/2)*Product(1+x^i*y^j), i, j>=0.
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EXAMPLE
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The second row (n=1) is 1,1 since (1,0) and (0,1) each have a single partition.
The third row (n=2) is 1, 2, 1 from (2,0), (1,1) or (1,0)+(0,1), (0,2).
In the fourth row, T(1,3)=5 from (1,3), (0,3)+(1,0), (0,2)+(1,1), (0,2)+(0,1)+(1,0), (0,1)+(1,2).
The triangle begins:
1;
1, 1;
1, 2, 1;
2, 3, 3, 2;
2, 5, 5, 5, 2;
3, 7, 9, 9, 7, 3;
4, 10, 14, 17, 14, 10, 4;
5, 14, 21, 27, 27, 21, 14, 5;
6, 19, 31, 42, 46, 42, 31, 19, 6;
8, 25, 44, 64, 74, 74, 64, 44, 25, 8;
...
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MATHEMATICA
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max = 10; f[x_, y_] := Product[1 + x^n*y^k, {n, 0, max}, {k, 0, max}]/2; se = Series[f[x, y], {x, 0, max}, {y, 0, max}] ; coes = CoefficientList[ se, {x, y}]; t[n_, k_] := coes[[n-k+1, k+1]]; Flatten[ Table[ t[n, k], {n, 0, max}, {k, 0, n}]] (* Jean-François Alcover, Dec 06 2011 *)
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PROG
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(Haskell) see Zumkeller link.
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CROSSREFS
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See A201377 for the same triangle formatted in a different way.
The outer diagonals are T(n,0) = T(n,n) = A000009(n).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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