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A000412
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Number of bipartite partitions of n white objects and 3 black ones.
(Formerly M2657 N1060)
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6
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3, 7, 16, 31, 57, 97, 162, 257, 401, 608, 907, 1325, 1914, 2719, 3824, 5313, 7316, 9973, 13495, 18105, 24132, 31938, 42021, 54948, 71484, 92492, 119120, 152686, 194887, 247693, 313613, 395547, 497154, 622688, 777424, 967525, 1200572, 1485393, 1832779, 2255317
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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^3 where p and q are distinct primes.
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
a(n) is the number of multiset partitions of the multiset {r^n, s^3}. - 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) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Feb 01 2016
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MATHEMATICA
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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 *)
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 *)
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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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