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A002755 Number of bipartite partitions of n white objects and 6 black ones.
(Formerly M4784 N2041)
21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
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
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 *)
PROG
(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
CROSSREFS
Column 6 of A054225.
Cf. A005380.
Sequence in context: A109943 A303856 A137411 * A346852 A157827 A242276
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Christian G. Bower, Jan 08 2004
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)