A000491 Number of bipartite partitions of n white objects and 5 black ones. (Formerly M4365 N1830)

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

0,1

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

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)

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

Adjacent sequences: A000488 A000489 A000490 * A000492 A000493 A000494

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by Christian G. Bower, Jan 08 2004

Status

approved