A000491 Number of bipartite partitions of n white objects and 5 black ones.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• F. C. Auluck, On partitions of bipartite numbers, Proc. Cambridge Philos. Soc. 49, (1953). 72-83.
• F. C. Auluck, On partitions of bipartite numbers, annotated scan of a few pages.
• 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)

Crossrefs

Column 5 of A054225.

Cf. A005380.

Programs

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 *)

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

Extensions

Edited by Christian G. Bower, Jan 08 2004