A002755 Number of bipartite partitions of n white objects and 6 black ones.

11, 30, 77, 162, 323, 589, 1043, 1752, 2876, 4571, 7128, 10860, 16306, 24051, 35040, 50355, 71609, 100697, 140349, 193784, 265505, 360889, 487214, 653243, 870613, 1153322, 1519658, 1991689, 2597762, 3372107, 4358198, 5608418, 7188632

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Number of ways to factor p^n*q^6 where p and q are distinct primes.

a(n) is the number of multiset partitions of the multiset {r^n, s^6}. - 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
• 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 6 of A054225.

Cf. A005380.

Programs

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[3^6*2^n, 3^6*2^n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[(11 + 8*x + 6*x^2 - 7*x^4 - 13*x^5 - 19*x^6 - 10*x^7 - 3*x^8 + 7*x^9 + 11*x^10 + 15*x^11 + 6*x^12 - 2*x^14 - 7*x^15 - 4*x^16 - 2*x^17 + 3*x^18 + 2*x^19 - x^20)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 01 2016 *)

Python

from sympy import divisors, isprime
from functools import cache
@cache
def T(n, m): # after Indranil Ghosh in A001055
    if isprime(n): return 1 if n <= m else 0
    s = sum(T(n//d, d) for d in divisors(n)[1:-1] if d <= m)
    return s + 1 if n <= m else s
def a(n): return (lambda x: T(x, x))(2**n * 3**6)
print([a(n) for n in range(33)]) # Michael S. Branicky, Nov 30 2021

Formula

a(n) = if n <= 6 then A054225(6,n) else A054225(n,6). - Reinhard Zumkeller, Nov 30 2011

a(n) ~ sqrt(3) * n^2 * exp(Pi*sqrt(2*n/3)) / (40*Pi^6). - Vaclav Kotesovec, Feb 01 2016

Extensions

Edited by Christian G. Bower, Jan 08 2004