%I M5123 N2219 #41 Jan 01 2024 08:01:16
%S 22,67,181,401,831,1576,2876,4987,8406,13715,21893,34134,52327,78785,
%T 116982,171259,247826,354482,502090,704265,979528,1351109,1849932,
%U 2514723,3396262,4557867,6081466,8068930,10650479,13987419,18283999
%N Number of bipartite partitions of n white objects and 8 black ones.
%C Number of ways to factor p^n*q^8 where p and q are distinct primes.
%C a(n) is the number of multiset partitions of the multiset {r^n, s^8}. - _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="/A002757/b002757.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 <= 8 then A054225(8,n) else A054225(n,8). - _Reinhard Zumkeller_, Nov 30 2011
%F a(n) ~ 3*sqrt(3) * n^3 * exp(Pi*sqrt(2*n/3)) / (1120*Pi^8). - _Vaclav Kotesovec_, Feb 01 2016
%t p = 2; q = 3; 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[p^n*q^8, p^n*q^8]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)
%t nmax = 50; CoefficientList[Series[(22 + 23*x + 25*x^2 + 16*x^3 + 4*x^4 - 14*x^5 - 34*x^6 - 50*x^7 - 65*x^8 - 52*x^9 - 32*x^10 + 5*x^11 + 27*x^12 + 57*x^13 + 67*x^14 + 65*x^15 + 42*x^16 + 15*x^17 - 14*x^18 - 34*x^19 - 40*x^20 - 46*x^21 - 26*x^22 - 8*x^23 + 8*x^24 + 11*x^25 + 18*x^26 + 14*x^27 + 9*x^28 + 3*x^29 - 7*x^30 - 7*x^31 - 6*x^32 + 3*x^33 + 3*x^34 - x^35)/((1-x) * (1-x^2) * (1-x^3) * (1-x^4) * (1-x^5) * (1-x^6) * (1-x^7) * (1-x^8)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Feb 01 2016 *)
%Y Column 8 of A054225.
%Y Cf. A005380.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Edited by _Christian G. Bower_, Jan 08 2004