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 A002774 Number of bipartite partitions of n white objects and n black ones. (Formerly M1925 N0760) 11

%I M1925 N0760

%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.

%C a(n) = A054225(n,n). - _Reinhard Zumkeller_, Nov 30 2011

%D F. C. Auluck, On partitions of bipartite numbers. Proc. Cambridge Philos. Soc. 49, (1953). 72-83.

%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 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, <a href="/A002774/b002774.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = A054225(2n, n) = A091437(2n).

%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

%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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Last modified May 18 20:27 EDT 2013. Contains 225428 sequences.