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A054242 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. 13
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; text; internal format)
OFFSET
0,5
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.
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
LINKS
S. M. Luthra, Partitions of bipartite numbers when the summands are unequal, Proceedings of the Indian National Science Academy, vol.23, 1957, issue 5A, p. 370-376. [broken link]
FORMULA
G.f.: (1/2)*Product(1+x^i*y^j), i, j>=0.
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;
...
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}]] (* Jean-François Alcover, Dec 06 2011 *)
PROG
(Haskell) see Zumkeller link.
CROSSREFS
See A201377 for the same triangle formatted in a different way.
The outer diagonals are T(n,0) = T(n,n) = A000009(n).
Cf. A054225.
T(2*n,n) = A219554(n). Row sums give A219555. - Alois P. Heinz, Nov 22 2012
Sequence in context: A232094 A143902 A085472 * A033767 A033775 A033791
KEYWORD
easy,nonn,tabl,nice
AUTHOR
Marc LeBrun, Feb 08 2000 and Jul 01 2003
EXTENSIONS
Entry revised by N. J. A. Sloane, Nov 30 2011, to incorporate corrections provided by Reinhard Zumkeller, who also contributed the alternative version A201377.
STATUS
approved

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Last modified July 18 18:59 EDT 2024. Contains 374388 sequences. (Running on oeis4.)