A219554 Number of bipartite partitions of (n,n) into distinct pairs.

1, 2, 5, 17, 46, 123, 323, 809, 1966, 4660, 10792, 24447, 54344, 118681, 254991, 539852, 1127279, 2323849, 4733680, 9535079, 19005282, 37507802, 73333494, 142112402, 273092320, 520612305, 984944052, 1849920722, 3450476080, 6393203741, 11770416313, 21538246251

Offset: 0

Author

Alois P. Heinz, Nov 22 2012

Keywords

• nonn
• nice

Comments

Number of factorizations of p^n*q^n into distinct factors where p, q are distinct primes.

From Vaclav Kotesovec, Feb 05 2016: (Start)

Formula (15) in the article by S. M. Luthra: "Partitions of bipartite numbers when the summands are unequal", p. 376, is incorrect. The similar error is also in the article by F. C. Auluck: "On partitions of bipartite numbers" (see A002774).

The correct formula (15) is q(m, n) ~ c/(2*sqrt(3)*Pi) * exp(3*c*(m*n)^(1/3) + 3*d*(m^(2/3)/n^(1/3) + n^(2/3)/m^(1/3)) - 3*log(2)/4 + (m/n + n/m)*log(2)/12 + 3*d^2/c - 3*d^2*(m/n + n/m)/c - 2*log(m*n)/3), where m and n are of the same order, c = (3/4*Zeta(3))^(1/3), d = Zeta(2)/(12*c).

If m = n then q(m,n) = a(n).

For the asymptotic formula for fixed m see A054242.

(End)

Examples

			a(0) = 1: [].
a(1) = 2: [(1,1)], [(1,0),(0,1)].
a(2) = 5: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(1,2),(1,0)], [(1,1),(1,0),(0,1)].
		

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..100 from Alois P. Heinz)
• 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]

Crossrefs

Column k=2 of A219585.

Cf. A002774, A219560, A219561, A219565.

Cf. A054242, A000009, A036469, A268345, A268346, A268347, A268348.

Programs

Mathematica

(* This program is not convenient for a large number of terms *)
a[n_] := If[n == 0, 1, (1/2) Coefficient[Product[O[x]^(n+1) + O[y]^(n+1) + (1 + x^i y^j ), {i, 0, n}, {j, 0, n}] // Normal, (x y)^n]];
a /@ Range[0, 31] (* Jean-François Alcover, Jun 26 2013, updated Sep 16 2019 *)
nmax = 20; p = 1; Do[Do[p = Expand[p*(1 + x^i*y^j)]; If[i*j != 0, p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &]], {i, 0, nmax}], {j, 0, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Flatten[{1, Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}]}] (* Vaclav Kotesovec, Jan 15 2016 *)

Formula

a(n) = [x^n*y^n] 1/2 * Product_{i,j>=0} (1+x^i*y^j).

a(n) = A054242(2*n,n) = A201377(n,n).

a(n) ~ Zeta(3)^(1/3) * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3) + Pi^2 * n^(1/3) / (6^(4/3) * Zeta(3)^(1/3)) - Pi^4/(1296*Zeta(3))) / (2^(9/4) * 3^(1/6) * Pi * n^(4/3)). - Vaclav Kotesovec, Jan 31 2016