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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-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 *) 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.)