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 A000491 Number of bipartite partitions of n white objects and 5 black ones. (Formerly M4365 N1830) 5
 7, 19, 47, 97, 189, 339, 589, 975, 1576, 2472, 3804, 5727, 8498, 12400, 17874, 25433, 35818, 49908, 68939, 94378, 128234, 172917, 231630, 308240, 407804, 536412, 701910, 913773, 1184022, 1527165, 1961432, 2508762, 3196473, 4057403, 5132066 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Number of ways to factor p^n*q^5 where p and q are distinct primes. a(n) = if n <= 5 then A054225(5,n) else A054225(n,5). [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..500 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) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - Vaclav Kotesovec, Feb 01 2016 MAPLE with(numtheory): b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0,       add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))     end: a:= n-> b(243*2^n\$2): seq(a(n), n=0..40);  # Alois P. Heinz, Jun 27 2013 MATHEMATICA 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 *) 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 *) CROSSREFS Column 5 of A054225. Cf. A005380. Sequence in context: A238730 A139865 A146403 * A097039 A067651 A090025 Adjacent sequences:  A000488 A000489 A000490 * A000492 A000493 A000494 KEYWORD nonn AUTHOR EXTENSIONS Edited by Christian G. Bower, Jan 08 2004 STATUS approved

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