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A000465
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Number of bipartite partitions of n white objects and 4 black ones.
(Formerly M3821 N1565)
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5
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5, 12, 29, 57, 109, 189, 323, 522, 831, 1279, 1941, 2876, 4215, 6066, 8644, 12151, 16933, 23336, 31921, 43264, 58250, 77825, 103362, 136371, 178975, 233532, 303268, 391831, 504069, 645520, 823419, 1046067, 1324136, 1669950, 2099104, 2629685, 3284325, 4089300
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OFFSET
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0,1
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COMMENTS
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Number of ways to factor p^n*q^4 where p and q are distinct primes.
a(n) is the number of multiset partitions of the multiset {r^n, s^4}. - Joerg Arndt, Jan 01 2024
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REFERENCES
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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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) ~ sqrt(3) * n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Feb 01 2016
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MATHEMATICA
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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 *)
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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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