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 A000412 Number of bipartite partitions of n white objects and 3 black ones. (Formerly M2657 N1060) 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Number of ways to factor p^n*q^3 where p and q are distinct primes. a(n) = if n <= 3 then A054225(3,n) else A054225(n,3). [Reinhard Zumkeller, Nov 30 2011] REFERENCES 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). LINKS Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..100 from Alois P. Heinz) F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83. 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 (Annotated scanned pages from, plus a review) FORMULA a(n) ~ exp(Pi*sqrt(2*n/3)) * sqrt(n) / (2*sqrt(2)*Pi^3). - Vaclav Kotesovec, Feb 01 2016 MATHEMATICA 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 *) CROSSREFS Column 3 of A054225. Cf. A005380. Sequence in context: A184677 A224340 A240739 * A192964 A179904 A161810 Adjacent sequences:  A000409 A000410 A000411 * A000413 A000414 A000415 KEYWORD nonn AUTHOR EXTENSIONS Edited by Christian G. Bower, Jan 08 2004 STATUS approved

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