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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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6
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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.
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LINKS
| Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377
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FORMULA
| G.f.: (1/2)*Product(1+x^i*y^j), i, j>=0.
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EXAMPLE
| 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
| 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}]] (* From Jean-François Alcover, Dec 06 2011 *)
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PROG
| (Haskell) see Zumkeller link.
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CROSSREFS
| See A201377 for the same triangle formatted in a different way.
The outer diagonals are T(n,0) = T(0,n) = A000009(n).
Cf. A054225.
Sequence in context: A058063 A143902 A085472 * A033767 A033775 A033791
Adjacent sequences: A054239 A054240 A054241 * A054243 A054244 A054245
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KEYWORD
| easy,nonn,tabl,nice
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AUTHOR
| Marc LeBrun (mlb(AT)well.com), Feb 08 2000 and Jul 01 2003
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EXTENSIONS
| Entry revised by N. J. A. Sloane, Nov 30 2011, to incorporate corrections provided by Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), who also contributed the alternative version A201377.
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