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%I M1168 N0447 #69 Jan 01 2024 08:00:55
%S 2,4,9,16,29,47,77,118,181,267,392,560,797,1111,1541,2106,2863,3846,
%T 5142,6808,8973,11733,15275,19753,25443,32582,41569,52770,66757,84078,
%U 105555,131995,164566,204450,253292,312799,385285,473183,579722,708353,863553
%N Number of bipartite partitions of n white objects and 2 black ones.
%C Number of ways to factor p^n*q^2 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^2}. - _Joerg Arndt_, Jan 01 2024
%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 Amarnath Murthy, "Generalization of Smarandache Factor Partition introducing Smarandache Factor Partition". Smarandache Notions Journal, 1-2-3, vol. 11, 2000.
%D Amarnath Murthy, Program for finding out the number of Smarandache Factor Partitions. Smarandache Notions Journal, Vol. 13, 2002.
%D Amarnath Murthy, e-book, MS LIT format, "Ideas on Smarandache Notions".
%D Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.9, 1.14.
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Table A-1, page 778. - _N. J. A. Sloane_, Dec 30 2018
%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="/A000291/b000291.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 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 M. S. Cheema and H. Gupta, <a href="/A002755/a002755.pdf">Tables of Partitions of Gaussian Integers</a>, National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956. (Annotated scanned pages from, plus a review)
%F a(n) = if n <= 2 then A054225(2,n) else A054225(n,2). - _Reinhard Zumkeller_, Nov 30 2011
%F From _Vaclav Kotesovec_, Feb 01 2016, corrected Nov 05 2016: (Start)
%F a(n) = A000070(n) + A000097(n).
%F a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 83*Pi/(24*sqrt(6*n))).
%F (End)
%e a(2) = 9: let p = 2 and q = 3, p^2*q^2 = 36; there are 9 factorizations: (36), (18*2), (12*3), (9*4), (9*2^2), (6*6), (6*3*2), (4*3^2), (3^2*2^2).
%t max = 40; col = 2; s1 = Series[Product[1/(1-x^(n-k)*y^k), {n, 1, max+2}, {k, 0, n}], {y, 0, col}] // Normal; s2 = Series[s1, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s2, {x, 0, n}, {y, 0, col}]; Table[ a[n] , {n, 0, max}] (* _Jean-François Alcover_, Mar 13 2014 *)
%t nmax = 50; CoefficientList[Series[1/(1-x)*(1 + 1/(1-x^2))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)
%Y Column 2 of A054225.
%Y Cf. A005380.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Jan 08 2004
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