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%I M1925 N0760 #97 Feb 09 2024 12:37:11
%S 1,2,9,31,109,339,1043,2998,8406,22652,59521,151958,379693,927622,
%T 2224235,5236586,12130780,27669593,62229990,138095696,302673029,
%U 655627975,1404599867,2977831389,6251060785,12999299705,26791990052,54750235190,110977389012
%N Number of bipartite partitions of n white objects and n black ones.
%C Number of ways to factor p^n * q^n where p and q are distinct primes.
%D M. S. Cheema and H. Gupta, Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956, p. 1.
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, see p(n,n), page 778. - _N. J. A. Sloane_, Dec 30 2018
%D A. Murthy, Generalization of partition function, introducing Smarandache factor partitions. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
%D A. Murthy, Program for finding out the number of Smarandache factor partitions. (To be published in Smarandache Notions Journal).
%D Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.14.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz and Vaclav Kotesovec, <a href="/A002774/b002774.txt">Table of n, a(n) for n = 0..400</a> (terms 0..100 from Alois P. Heinz)
%H F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100028061">On partitions of bipartite numbers</a>, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
%H F. C. Auluck, <a href="/A002774/a002774.pdf">On partitions of bipartite numbers, annotated scan of a few pages.</a>
%H F. C. Auluck, <a href="/A002774/a002774_1.pdf">On partitions of bipartite numbers</a>, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 49, Issue 01, January 1953, pp. 72-83. (full article)
%H Katherine Ormeño Bastías, Paul Martin, and Steen Ryom-Hansen, <a href="https://arxiv.org/abs/2402.01890">On the spherical partition algebra</a>, arXiv:2402.01890 [math.RT], 2024.
%H M. S. Cheema and H. Gupta, <a href="/A002755/a002755.pdf">Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956</a> (Annotated scanned pages from, plus a review)
%F a(n) = A054225(2n, n) = A091437(2n).
%F a(n) ~ Zeta(3)^(19/36) * exp(3*Zeta(3)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (6*Zeta(3)^(1/3)) + Zeta'(-1) - Pi^4/(432*Zeta(3))) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)). - _Vaclav Kotesovec_, Jan 30 2016
%F Formula (25) in the article by Auluck is incorrect. The correct formula is: p(n,n) ~ c^(19/12) * exp(3*c*n^(2/3) + 3*d*n^(1/3) + Zeta'(-1) - 3*d^2/(4*c)) / (sqrt(3) * (2*Pi)^(3/2) * n^(55/36)), where c = Zeta(3)^(1/3), d = Zeta(2)/(3*c). Also formula (24) is incorrect. - _Vaclav Kotesovec_, Jan 30 2016
%F From _Vaclav Kotesovec_, Feb 04 2016: (Start)
%F The correct formula (24) is p(m,n) ~ c^(7/4)/(2*Pi*sqrt(3)) * exp(3*c*(m*n)^(1/3) + 3*d*(m+n)/(2*(m*n)^(1/3)) - 19*log(m*n)/24 - ((m/n - 2*n/m)*log(m) + (n/m - 2*m/n)*log(n))/36 - (m/n + n/m)*(log(c)/12 + Zeta'(-1) - 1/12 + 3*d^2/(4*c)) + 3*d^2/(4*c) - 3*log(2*Pi)/4 + fi((n/m)^(1/2))),
%F where m and n are of the same order, c = Zeta(3)^(1/3), d = Zeta(2)/(3*c) and fi(alfa) = Integral_{t=0..infinity} (1/t)*(1/(exp(alfa*t)-1)/(exp(t/alfa)-1) - (alfa/t)/(exp(alfa*t)-1) - ((1/alfa)/t)/(exp(t/alfa)-1) + 1/t^2 + (1/4)/(exp(alfa*t)-1) + (1/4)/(exp(t/alfa)-1) - (alfa/4)/t - ((1/4)/alfa)/t).
%F If m = n then alfa = 1 and fi(1) = 3*Zeta'(-1) + log(2*Pi)/4 - 1/6.
%F For the asymptotic formula for fixed m see A054225.
%F (End)
%p with(numtheory):
%p b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0,
%p add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
%p end:
%p a:= n-> b(6^n$2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 27 2013
%t max = 26; se = Series[ Sum[ Log[1 - x^(n-k)*y^k], {n, 1, 2max}, {k, 0, n}], {x, 0, 2max}, {y, 0, 2max}]; coes = CoefficientList[ Series[ Exp[-se], {x, 0, 2max}, {y, 0, 2max}], {x, y}]; a[n_] := coes[[n+1, n+1]]; Table[a[n], {n, 0, max} ](* _Jean-François Alcover_, Dec 06 2011 *)
%Y Cf. A005380.
%Y Cf. A219554. Column k=2 of A219727. - _Alois P. Heinz_, Nov 26 2012
%Y Cf. also A000041, A000070, A000291, A000412, A000465, A000491, A002755, A002756, A002757, A002758, A002759, A277239.
%Y Main diagonal of A054225 if that entry is drawn as a square array. - _N. J. A. Sloane_, Dec 30 2018
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E Corrected using A000491.
%E Edited by _Christian G. Bower_, Jan 08 2004
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