%I M4784 N2041 #47 Jan 01 2024 08:01:26
%S 11,30,77,162,323,589,1043,1752,2876,4571,7128,10860,16306,24051,
%T 35040,50355,71609,100697,140349,193784,265505,360889,487214,653243,
%U 870613,1153322,1519658,1991689,2597762,3372107,4358198,5608418,7188632
%N Number of bipartite partitions of n white objects and 6 black ones.
%C Number of ways to factor p^n*q^6 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^6}. - _Joerg Arndt_, Jan 01 2024
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002755/b002755.txt">Table of n, a(n) for n = 0..1000</a>
%H M. S. Cheema and H. Gupta, <a href="/A002755/a002755.pdf">Tables of Partitions of Gaussian Integers. National Institute of Sciences of India, Mathematical Tables, Vol. 1, New Delhi, 1956</a> (Annotated scanned pages from, plus a review)
%F a(n) = if n <= 6 then A054225(6,n) else A054225(n,6). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ sqrt(3) * n^2 * exp(Pi*sqrt(2*n/3)) / (40*Pi^6). - _Vaclav Kotesovec_, Feb 01 2016
%t 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_ *)
%t 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 *)
%o (Python)
%o from sympy import divisors, isprime
%o from functools import cache
%o @cache
%o def T(n, m): # after Indranil Ghosh in A001055
%o if isprime(n): return 1 if n <= m else 0
%o s = sum(T(n//d, d) for d in divisors(n)[1:-1] if d <= m)
%o return s + 1 if n <= m else s
%o def a(n): return (lambda x: T(x, x))(2**n * 3**6)
%o print([a(n) for n in range(33)]) # _Michael S. Branicky_, Nov 30 2021
%Y Column 6 of A054225.
%Y Cf. A005380.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Jan 08 2004