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A219554 Number of bipartite partitions of (n,n) into distinct pairs. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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)

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]

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

EXAMPLE

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)].

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-Fran├žois 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 *)

CROSSREFS

Column k=2 of A219585.

Cf. A002774, A219560, A219561, A219565.

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

Sequence in context: A096295 A215580 A275210 * A074494 A051438 A148401

Adjacent sequences:  A219551 A219552 A219553 * A219555 A219556 A219557

KEYWORD

nonn,nice

AUTHOR

Alois P. Heinz, Nov 22 2012

STATUS

approved

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Last modified January 25 11:49 EST 2022. Contains 350567 sequences. (Running on oeis4.)