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%I M4365 N1830 #53 Jan 01 2024 08:01:03
%S 7,19,47,97,189,339,589,975,1576,2472,3804,5727,8498,12400,17874,
%T 25433,35818,49908,68939,94378,128234,172917,231630,308240,407804,
%U 536412,701910,913773,1184022,1527165,1961432,2508762,3196473,4057403,5132066
%N Number of bipartite partitions of n white objects and 5 black ones.
%C Number of ways to factor p^n*q^5 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^5}. - _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="/A000491/b000491.txt">Table of n, a(n) for n = 0..1000</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 <= 5 then A054225(5,n) else A054225(n,5). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - _Vaclav Kotesovec_, Feb 01 2016
%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(243*2^n$2):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 27 2013
%t b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, DeleteCases[Divisors[n], 1|n]}]]; a[n_] := b[3^5*2^n, 3^5*2^n]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 13 2014, after _Alois P. Heinz_ *)
%t nmax = 50; CoefficientList[Series[(7 + 5*x + 2*x^2 - 2*x^3 - 7*x^4 - 9*x^5 - 6*x^6 + x^7 + 4*x^8 + 6*x^9 + 3*x^10 + x^11 - 3*x^12 - 2*x^13 + x^14)/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)
%Y Column 5 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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