

A054242


Triangle read by rows: row n (n>=0) gives the number of partitions of (n,0), (n1,1), (n2,2), ..., (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
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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



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[[nk+1, k+1]]; Flatten[ Table[ t[n, k], {n, 0, max}, {k, 0, n}]] (* JeanFrançois 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).


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



