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%I M2657 N1060 #91 Jan 01 2024 08:00:59
%S 3,7,16,31,57,97,162,257,401,608,907,1325,1914,2719,3824,5313,7316,
%T 9973,13495,18105,24132,31938,42021,54948,71484,92492,119120,152686,
%U 194887,247693,313613,395547,497154,622688,777424,967525,1200572,1485393,1832779,2255317
%N Number of bipartite partitions of n white objects and 3 black ones.
%C Number of ways to factor p^n*q^3 where p and q are distinct primes.
%C Number of Gaussian partitions of n+3*i or 3+n*i where a "Gaussian partition" is a way of writing a Gaussian integer with nonnegative parts as a sum of Gaussian integers with nonnegative parts, imaginary numbers and real numbers. For k = 3+1*i (where i is the imaginary unit), the a(1)=7 ways to write k (where parentheses represent a complex number and a lack of them represents a sum of a real and imaginary number) would be 3+i, (3+i), 2+1+i, (2+i)+1, (1+i)+2, 1+1+1+i, (1+i)+1+1. - _Yali Harrary_, Nov 20 2022
%C a(n) is the number of multiset partitions of the multiset {r^n, s^3}. - _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 Vaclav Kotesovec, <a href="/A000412/b000412.txt">Table of n, a(n) for n = 0..10000</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), pp. 72-83.
%H F. C. Auluck, <a href="/A002774/a002774.pdf">On partitions of bipartite numbers</a>, annotated scan of a few pages.
%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 <= 3 then A054225(3,n), otherwise a(n) = A054225(n,3). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (2*sqrt(2)*Pi^3). - _Vaclav Kotesovec_, Feb 01 2016
%F a(n) = A000098(n) + A000070(n) + A014153(n). - _Yali Harrary_, Nov 20 2022
%t max = 40; col = 3; 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[(3 + x - x^2 - 2*x^3 - x^4 + x^5)/((1-x)*(1-x^2)*(1-x^3)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)
%Y Column 3 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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