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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) is the number of multiset partitions of the multiset {r^n, s^5}. - Joerg Arndt, Jan 01 2024
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
F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
FORMULA
a(n) = if n <= 5 then A054225(5,n) else A054225(n,5). - Reinhard Zumkeller, Nov 30 2011
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 A357301
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Christian G. Bower, Jan 08 2004
STATUS
approved

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Last modified April 22 14:29 EDT 2024. Contains 371904 sequences. (Running on oeis4.)