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.

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

Offset: 0

Author

Marc LeBrun, Feb 08 2000 and Jul 01 2003

Keywords

• easy
• nonn
• tabl
• nice

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

Examples

			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;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..75, flattened
• 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]
• Reinhard Zumkeller, Haskell programs for A054225, A054242, A201376, A201377

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

Columns 0-5: A000009, A036469, A268345, A268346, A268347, A268348.

Programs

Haskell

see Zumkeller link.

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 *)

Formula

G.f.: (1/2)*Product(1+x^i*y^j), i, j>=0.

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.