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%I M3821 N1565 #50 Jan 01 2024 08:01:35
%S 5,12,29,57,109,189,323,522,831,1279,1941,2876,4215,6066,8644,12151,
%T 16933,23336,31921,43264,58250,77825,103362,136371,178975,233532,
%U 303268,391831,504069,645520,823419,1046067,1324136,1669950,2099104,2629685,3284325,4089300
%N Number of bipartite partitions of n white objects and 4 black ones.
%C Number of ways to factor p^n*q^4 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^4}. - _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 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="/A000465/b000465.txt">Table of n, a(n) for n = 0..5000</a>
%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. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956</a> (Annotated scanned pages from, plus a review)
%F a(n) = if n <= 4 then A054225(4,n) else A054225(n,4). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ sqrt(3) * n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - _Vaclav Kotesovec_, Feb 01 2016
%t max = 40; col = 4; 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[(5 + 2*x - 3*x^3 - 5*x^4 - x^5 + 3*x^7 + x^8 - x^9)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)
%Y Column 4 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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