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 A000465 Number of bipartite partitions of n white objects and 4 black ones. (Formerly M3821 N1565) 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Number of ways to factor p^n*q^4 where p and q are distinct primes. a(n) = if n <= 4 then A054225(4,n) else A054225(n,4). [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, Table of n, a(n) for n = 0..100 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) ~ sqrt(3) * n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Feb 01 2016 MATHEMATICA 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 *) CROSSREFS Column 4 of A054225. Cf. A005380. Sequence in context: A002767 A055245 A196410 * A283506 A266471 A069306 Adjacent sequences:  A000462 A000463 A000464 * A000466 A000467 A000468 KEYWORD nonn AUTHOR EXTENSIONS Edited by Christian G. Bower, Jan 08 2004 STATUS approved

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