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A002755
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Number of bipartite partitions of n white objects and 6 black ones.
(Formerly M4784 N2041)
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21
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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a(n) ~ sqrt(3) * n^2 * exp(Pi*sqrt(2*n/3)) / (40*Pi^6). - Vaclav Kotesovec, Feb 01 2016
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MATHEMATICA
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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 *)
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PROG
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(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)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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