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%I #61 Aug 17 2020 01:58:20
%S 1,2,5,17,46,123,323,809,1966,4660,10792,24447,54344,118681,254991,
%T 539852,1127279,2323849,4733680,9535079,19005282,37507802,73333494,
%U 142112402,273092320,520612305,984944052,1849920722,3450476080,6393203741,11770416313,21538246251
%N Number of bipartite partitions of (n,n) into distinct pairs.
%C Number of factorizations of p^n*q^n into distinct factors where p, q are distinct primes.
%C From _Vaclav Kotesovec_, Feb 05 2016: (Start)
%C 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).
%C 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).
%C If m = n then q(m,n) = a(n).
%C For the asymptotic formula for fixed m see A054242.
%C (End)
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A219554/b219554.txt">Table of n, a(n) for n = 0..400</a> (terms 0..100 from Alois P. Heinz)
%H S. M. Luthra, <a href="http://www.insa.nic.in/writereaddata/UpLoadedFiles/PINSA/Vol23A_1957_5_Art04.pdf">Partitions of bipartite numbers when the summands are unequal</a>, Proceedings of the Indian National Science Academy, vol. 23, 1957, issue 5A, p. 370-376. [broken link]
%F a(n) = [x^n*y^n] 1/2 * Product_{i,j>=0} (1+x^i*y^j).
%F a(n) = A054242(2*n,n) = A201377(n,n).
%F 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
%e a(0) = 1: [].
%e a(1) = 2: [(1,1)], [(1,0),(0,1)].
%e a(2) = 5: [(2,2)], [(2,1),(0,1)], [(2,0),(0,2)], [(1,2),(1,0)], [(1,1),(1,0),(0,1)].
%t (* This program is not convenient for a large number of terms *)
%t 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]];
%t a /@ Range[0, 31] (* _Jean-François Alcover_, Jun 26 2013, updated Sep 16 2019 *)
%t 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 *)
%Y Column k=2 of A219585.
%Y Cf. A002774, A219560, A219561, A219565.
%Y Cf. A054242, A000009, A036469, A268345, A268346, A268347, A268348.
%K nonn,nice
%O 0,2
%A _Alois P. Heinz_, Nov 22 2012
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